Numerical approximations for nonlinearstochastic partial differential equations
Arnulf Jentzen (ETH Zurich, Switzerland)Joint works with
Sonja Cox (University of Amsterdam, the Netherlands),Arnaud Debussche (ENS Rennes, France),
Giuseppe Da Prato (Scuola Normale Superiore di Pisa, Italy),Martin Hairer (University of Warwick, UK),
Mario Hefter (University of Kaiserslautern, Germany),Martin Hutzenthaler (University of Duisburg-Essen, Germany),
Ladislas Jacobe de Naurois (ETH Zurich, Switzerland),Thomas Müller-Gronbach (University of Passau, Germany),
Ryan Kurniawan (ETH Zurich, Switzerland),Michael Röckner (Bielefeld University, Germany),
Timo Welti (ETH Zurich, Switzerland), andLarisa Yaroslavtseva (University of Passau, Germany)
Workshop on Numerics for Stochastic Partial DifferentialEquations and their Applications, Linz, Austria
Monday, 10 December 2016
Some examples of SDEs
Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X
(2)0 > 0, ρ ∈ [−1, 1] and
∂∂t X
(1)t = α X
(1)t +
√X
(2)t X
(1)t
∂∂t W
(1)t
∂∂t X
(2)t = δ − γX
(2)t + β
√X
(2)t
(ρ ∂∂t W
(1)t +
√1− ρ2 ∂
∂t W(2)t
)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W
(1)t ,W
(2)t ))t∈[0,T ] is a two-dim. BM.
Nonlinear stochastic heat equation (Parabolic Anderson model)
∂∂t Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)
∂2
∂t2 Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation
∂∂t Xt(x) = ∂2
∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].
Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X
(2)0 > 0, ρ ∈ [−1, 1] and
∂∂t X
(1)t = α X
(1)t +
√X
(2)t X
(1)t
∂∂t W
(1)t
∂∂t X
(2)t = δ − γX
(2)t + β
√X
(2)t
(ρ ∂∂t W
(1)t +
√1− ρ2 ∂
∂t W(2)t
)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W
(1)t ,W
(2)t ))t∈[0,T ] is a two-dim. BM.
Nonlinear stochastic heat equation (Parabolic Anderson model)
∂∂t Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)
∂2
∂t2 Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation
∂∂t Xt(x) = ∂2
∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].
Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X
(2)0 > 0, ρ ∈ [−1, 1] and
∂∂t X
(1)t = α X
(1)t +
√X
(2)t X
(1)t
∂∂t W
(1)t
∂∂t X
(2)t = δ − γX
(2)t + β
√X
(2)t
(ρ ∂∂t W
(1)t +
√1− ρ2 ∂
∂t W(2)t
)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W
(1)t ,W
(2)t ))t∈[0,T ] is a two-dim. BM.
Nonlinear stochastic heat equation (Parabolic Anderson model)
∂∂t Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)
∂2
∂t2 Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation
∂∂t Xt(x) = ∂2
∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].
Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X
(2)0 > 0, ρ ∈ [−1, 1] and
∂∂t X
(1)t = α X
(1)t +
√X
(2)t X
(1)t
∂∂t W
(1)t
∂∂t X
(2)t = δ − γX
(2)t + β
√X
(2)t
(ρ ∂∂t W
(1)t +
√1− ρ2 ∂
∂t W(2)t
)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W
(1)t ,W
(2)t ))t∈[0,T ] is a two-dim. BM.
Nonlinear stochastic heat equation (Parabolic Anderson model)
∂∂t Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)
∂2
∂t2 Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation
∂∂t Xt(x) = ∂2
∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].
Heston model Consider α, γ ∈ R, β, δ, X(1)0 , X
(2)0 > 0, ρ ∈ [−1, 1] and
∂∂t X
(1)t = α X
(1)t +
√X
(2)t X
(1)t
∂∂t W
(1)t
∂∂t X
(2)t = δ − γX
(2)t + β
√X
(2)t
(ρ ∂∂t W
(1)t +
√1− ρ2 ∂
∂t W(2)t
)for t ∈ [0, T ], where (Wt)t∈[0,T ] = ((W
(1)t ,W
(2)t ))t∈[0,T ] is a two-dim. BM.
Nonlinear stochastic heat equation (Parabolic Anderson model)
∂∂t Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], where (Wt)t∈[0,T ] is acylindrical IdL2((0,1);R)-Wiener process and b : R→ R is regular.Nonlinear stochastic Wave equation (Hyperbolic Anderson model)
∂2
∂t2 Xt(x) = ∂2
∂x2 Xt(x) + b(Xt(x)) ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].Stochastic Burgers equation
∂∂t Xt(x) = ∂2
∂x2 Xt(x)− Xt(x) · ∂∂x Xt(x) + ∂∂t Wt(x)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ].
Some difficulties in the numerical approximations of SDEs
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Theorem (Hairer, Hutzenthaler & J 2015 AOP)
Let T ∈ (0,∞), d ∈ 4, 5, . . . , ξ ∈ Rd . Then there exist globally boundedµ, σ ∈ C∞(Rd ,Rd ) such that for every probability space (Ω,F ,P), every Brownianmotion W : [0, T ]× Ω→ R, every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every Y N : 0, 1, . . . ,N × Ω→ R4, N ∈ N, with∀N ∈ N, n ∈ 0, 1, . . . ,N − 1 : Y N
0 = X0 and
Y Nn+1 = Y N
n + µ(Y Nn ) T
N + σ(Y Nn )(
W (n+1)TN
−W nTN
)(Euler-Maruyama approximations) we have ∀α ∈ [0,∞) :
limN→∞
(Nα∥∥E[XT
]− E
[Y N
N
]∥∥) =
0 : α = 0
∞ : α > 0.
Plot of∥∥E[XT
]− E
[Y N
N
]∥∥ for T = 2 and N ∈ 21, 22, . . . , 230.
100
102
104
106
108
1010
10−12
10−10
10−8
10−6
10−4
10−2
100
Number N of time discretizations
Ap
pro
xim
atio
n e
rro
r o
f th
e m
ea
n
Approximation error of the mean
A function with order 0
Order line 1/2
Order line 1
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Dimension d ≥ 4: J, Müller-Gronbach & Yaroslavtseva 2016 CMS
Weak convergence and d ≥ 4: Müller-Gronbach & Yaroslavtseva 2016 SAA(to appear)
Adaptive approximations and d ≥ 4: Yaroslavtseva 2016
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
Theorem (Hefter & J 2016)
Let T , δ, β ∈ (0,∞), γ, ξ ∈ [0,∞), let (Ω,F ,P) be a probability space, letW : [0, T ]× Ω→ R be a Brownian motion, let X : [0, T ]× Ω→ R be a solution of
dXt = (δ − γXt) dt + β√
Xt dWt , t ∈ [0, T ], X0 = ξ.
Then there exists a c ∈ (0,∞) such that for all N ∈ N we have
infu : RN→Rmeasurable
E[∣∣XT − u(W T
N,W 2T
N, . . . ,WT )
∣∣] ≥ c · N−min1, 2δβ2 .
a
0 5 10 15 20 250
20
40
60
80
100
120
absolute
quantity
ofstocksin
theS&P500
2δ/β2 in the Heston model
The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ
β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.
More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1
10 .
a
0 5 10 15 20 250
20
40
60
80
100
120
absolute
quantity
ofstocksin
theS&P500
2δ/β2 in the Heston model
The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ
β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.
More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1
10 .
a
0 5 10 15 20 250
20
40
60
80
100
120
absolute
quantity
ofstocksin
theS&P500
2δ/β2 in the Heston model
The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ
β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.
More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1
10 .
a
0 5 10 15 20 250
20
40
60
80
100
120
absolute
quantity
ofstocksin
theS&P500
2δ/β2 in the Heston model
The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ
β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.
More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1
10 .
a
0 5 10 15 20 250
20
40
60
80
100
120
absolute
quantity
ofstocksin
theS&P500
2δ/β2 in the Heston model
The S&P 500 (the Standard & Poor’s 500) is a stock market index.In Hutzenthaler, J & Noll 2016 we calibrate 498 stocks from the S&P 500 within theHeston model: 359 stocks satisfy 2δ
β2 ≤ 25, 162 stocks (≈ 32%) satisfy 2δβ2 < 1.
More than 100 stocks (= 20%) satisfy 2δβ2 ≤ 1
10 .
Convergence results
Weak convergence results for SPDEs in the literature, e.g.,Hausenblas 2003 Progr. Probab.
Shardlow 2003 BIT
De Bouard & Debussche 2006 Appl. Math. Optim.
Debussche & Printemps 2007 arXiv (2009 Math. Comp.)Debussche 2008 arXiv (2011 Math. Comp.)Geissert, Kovacs & Larsson 2009 BIT
Lindner & Schilling 2009 arXiv (2013 Potential Anal.)Hausenblas 2011 J. Comput. Appl. Math.
Dörsek 2011 arXiv (2012 SIAM J. Numer. Anal.)Dörsek 2011 PhD thesis, TU Wien
Kovacs, Larsson & Lindgren 2012 BIT
Kovacs, Larsson & Lindgren 2012 arXiv (2013 BIT)Wang & Gan 2012 J. Math. Anal. Appl.
Bréhier 2012 arXiv (2014 Potential Anal.)Kruse 2012 PhD thesis, Bielefeld University (2014 Lecture Notes in Mathematics)Lindgren 2012 PhD thesis, Chalmers University of Technology and University of Gothenburg
Andersson & Larsson 2012 arXiv
Andersson, Kruse & Larsson 2013 arXiv ( 2016 SPDEs: Anal. and Comp.)Bréhier & Kopec 2013 arXiv
Wang 2013 arXiv . . .
use for Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) or Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) that b
is constant.
Weak convergence results for SPDEs in the literature, e.g.,Hausenblas 2003 Progr. Probab.
Shardlow 2003 BIT
De Bouard & Debussche 2006 Appl. Math. Optim.
Debussche & Printemps 2007 arXiv (2009 Math. Comp.)Debussche 2008 arXiv (2011 Math. Comp.)Geissert, Kovacs & Larsson 2009 BIT
Lindner & Schilling 2009 arXiv (2013 Potential Anal.)Hausenblas 2011 J. Comput. Appl. Math.
Dörsek 2011 arXiv (2012 SIAM J. Numer. Anal.)Dörsek 2011 PhD thesis, TU Wien
Kovacs, Larsson & Lindgren 2012 BIT
Kovacs, Larsson & Lindgren 2012 arXiv (2013 BIT)Wang & Gan 2012 J. Math. Anal. Appl.
Bréhier 2012 arXiv (2014 Potential Anal.)Kruse 2012 PhD thesis, Bielefeld University (2014 Lecture Notes in Mathematics)Lindgren 2012 PhD thesis, Chalmers University of Technology and University of Gothenburg
Andersson & Larsson 2012 arXiv
Andersson, Kruse & Larsson 2013 arXiv ( 2016 SPDEs: Anal. and Comp.)Bréhier & Kopec 2013 arXiv
Wang 2013 arXiv . . .
use for Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) or Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) that b
is constant.
Weak convergence results for SPDEs in the literature, e.g.,Hausenblas 2003 Progr. Probab.
Shardlow 2003 BIT
De Bouard & Debussche 2006 Appl. Math. Optim.
Debussche & Printemps 2007 arXiv (2009 Math. Comp.)Debussche 2008 arXiv (2011 Math. Comp.)Geissert, Kovacs & Larsson 2009 BIT
Lindner & Schilling 2009 arXiv (2013 Potential Anal.)Hausenblas 2011 J. Comput. Appl. Math.
Dörsek 2011 arXiv (2012 SIAM J. Numer. Anal.)Dörsek 2011 PhD thesis, TU Wien
Kovacs, Larsson & Lindgren 2012 BIT
Kovacs, Larsson & Lindgren 2012 arXiv (2013 BIT)Wang & Gan 2012 J. Math. Anal. Appl.
Bréhier 2012 arXiv (2014 Potential Anal.)Kruse 2012 PhD thesis, Bielefeld University (2014 Lecture Notes in Mathematics)Lindgren 2012 PhD thesis, Chalmers University of Technology and University of Gothenburg
Andersson & Larsson 2012 arXiv
Andersson, Kruse & Larsson 2013 arXiv ( 2016 SPDEs: Anal. and Comp.)Bréhier & Kopec 2013 arXiv
Wang 2013 arXiv . . .
use for Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) or Xt (x) = ∂2
∂x2 Xt (x) + b(Xt (x))Wt (x) that b
is constant.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Nonlinear stochastic heat equations – Hefter, J, & Kurniawan 2016)
Let T > 0, p ≥ 2, let f , b : R→ R and ϕ : Lp((0, 1);R)→ R be C4 with Lipschitzcontinuous and bounded derivatives, let ξ : (0, 1)→ R be measurable and bounded,let (Ω,F ,P) be a probability space, let (Wt)t∈[0,T ] be an IdL2((0,1);R)-cylindricalWiener process, let X : [0, T ]× Ω→ Lp((0, 1);R) be a continuous mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + f(Xt(x)) + b(Xt(x))Wt(x) (*)
with Xt(0) = Xt(1) = 0 for x ∈ (0, 1), t ∈ [0, T ], and for every N ∈ N letY N : 0, 1, . . . ,N × Ω→ Lp((0, 1);R) be a time discrete exponential Eulerapproximation for the SPDE (*) with time step size T/N. Then
∀ ε > 0 : ∃C ∈ R : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
N )]∣∣ ≤ C · N(ε−1/2)
∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :(E[‖XT − Y N
N ‖2H
])1/2 ≤ C · N(ε−1/4) .
In the case f ≡ 0, b ≡ 1 : ∀ p ∈ [2,∞) : ∃ϕ ∈ C∞b (Lp((0, 1);R),R) :
∃C > 0 : ∀N ∈ N :∣∣E[ϕ(XT )
]− E
[ϕ(Y N
T )]∣∣ ≥ C · N−1/2.
Similar results for linear-implicit Euler, spatial approximations, and higherdimensions.
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Hyperbolic Anderson model; Jacobe de Naurois, J & Welti 2015)
Let H = L2((0, 1);R), ϕ ∈ C2b (H,R), ξ = (ξ0, ξ1) ∈ H1
0 ((0, 1);R)× H,f ∈ C3
b ((0, 1)× R,R), for every n ∈ N let en(·) =√
2 sin(nπ(·)) ∈ H and forevery N ∈ N ∪ ∞ let PN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let
X N : [0, T ]× Ω→ PN(H) be a mild solution of
Xt(x) = ∂2
∂x2 Xt(x) + PN f(x, Xt(x)) + PN Xt(x) Wt(x)
with X0(x) = ξ0(x) and X0(x) = ξ1(x) for t ∈ [0, T ], x ∈ (0, 1). Then∀ ε > 0 : ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N(ε−1).
Rate can essentially not be improved.
Similar results for time discretizations (Cox, J & Welti 2016).
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Theorem (Stochastic Burgers equation – Debussche, J, & Welti 2016)
Let H = L2((0, 1);R), ξ ∈ H, ϕ ∈ C3b (H,R), κ ∈ R, for every n ∈ N let
en(·) =√
2 sin(nπ(·)) ∈ H, and for every N ∈ N ∪ ∞ letPN(·) =
∑Nn=1 〈en, ·〉H en ∈ L(H) and let X N : [0, T ]× Ω→ PN(H) be a
continuous mild solution of
X Nt (x) = ∂2
∂x2 X Nt (x) + PN
(κ · X N
t (x) · ∂∂x X Nt (x)
)+ PN Wt(x)
with X N0 (x) = (PNξ)(x) for t ∈ [0, T ], x ∈ (0, 1). Then ∃C ≥ 0 : ∀N ∈ N :∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
In the case κ = 0 it holds ∃ϕ ∈ C3b (H,R), c,C ∈ (0,∞) : ∀N ∈ N :
c · N−1 ≤∣∣E[ϕ(X∞T )
]− E
[ϕ(X N
T )]∣∣ ≤ C · N−1.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Cahn-Hilliard Cook equation driven by a standard Wiener process:
∂∂t Xt(x) =
[− ∂4
x Xt(x) + ∂2x
(Xt(x)3 − Xt(x)
)]dt + ∂
∂t Wt(x)
for x ∈ (0, 1) with Neumann and no-flux boundary conditions and regular initial value.Kovacs, Larsson, Mesforush 2011, in particular, implies∀α ∈ (0, 2), ε > 0 : ∃Cα,ε ≥ 0 : ∀ h > 0 : ∃Ωε,h ⊆ Ω:
P(Ωε,h) ≥ 1− ε and 1Ωε,h supt∈[0,T ]
∥∥Xt − Y ht
∥∥ ≤ Cα,ε hα
(semi strong convergence rate) and hence
limh0
E
[sup
t∈[0,T ]
∥∥Xt − Y ht
∥∥2
]= 0.
Hutzenthaler & J 2014: ∀α ∈ (0, 2), p > 0 : ∃Cα,p ≥ 0 : ∀N ∈ N :∥∥∥∥∥ supt∈[0,T ]
∥∥Xt − Y Nt
∥∥∥∥∥∥∥Lp(Ω;R)
≤ Cα,p N−α.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Theorem (Gerencsér, J, & Salimova 2016)
Let T ∈ (0,∞), d ∈ 2, 3, 4, . . . , ξ ∈ Rd , (aN)N∈N ⊆ R satisfylimN→∞ aN = 0. Then there exist globally bounded µ, σ ∈ C∞(Rd ,Rd ) such thatfor every probability space (Ω,F ,P), every Brownian motion W : [0, T ]× Ω→ R,every solution X : [0, T ]× Ω→ Rd of
dXt = µ(Xt) dt + σ(Xt) dWt , t ∈ [0, T ], X0 = ξ,
and every N ∈ N we have
infs1,...,sN∈[0,T ]
infu : RN→Rd
measurable
E[∥∥XT − u
(Ws1 , . . . ,WsN
)∥∥] ≥ aN .
Remark: ∀α ∈ (0, 12 ), ε > 0 : ∃Cα,ε ≥ 0 : ∀N ∈ N : ∃Ωε,N ⊆ Ω:
P(Ωε,N) ≥ 1− ε and 1Ωε,N supt∈[0,T ]
∥∥Xt − Y Nt
∥∥ ≤ Cα,ε N−α
Gyöngy (1998): ∀α ∈ (0, 12 ) : ∃Cα : Ω→ [0,∞) : ∀N ∈ N :
supt∈[0,T ]
‖Xt − Y Nt ‖ ≤ Cα N−α P-a.s.
Plot of∥∥E[XT
]− E
[Y N
N
]∥∥ for T = 2 and N ∈ 21, 22, . . . , 230.
100
102
104
106
108
1010
10−12
10−10
10−8
10−6
10−4
10−2
100
Number N of time discretizations
Ap
pro
xim
atio
n e
rro
r o
f th
e m
ea
n
Approximation error of the mean
A function with order 0
Order line 1/2
Order line 1
Methods of proof
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Corollary (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let V be a separable Hilbert space and let ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 ≤ t .Then it holds P-a.s. that
ϕ(Xt) = ϕ(eA(t−t0)Xt0 ) +
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)F(Xs) ds
+
∫ t
t0
ϕ′(eA(t−s)Xs) eA(t−s)B(Xs) dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(eA(t−s)Xs)(
eA(t−s)B(Xs)u, eA(t−s)B(Xs)u)
ds.
Mild Itô formula shows
E[ϕ(Xt)
]= E
[ϕ(eAt X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)Xs)
(eA(t−s)B(Xs)u, eA(t−s)B(Xs)u
)]ds,
E[ϕ(PN(Xt))
]= E
[ϕ(eAt PN(X0)
]+ 1
2
∑u∈U
t∫0E[ϕ′′(eA(t−s)PN(Xs))
(eA(t−s)PNB(Xs)u, eA(t−s)PNB(Xs)u
)]ds.
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Definition (Mild Itô process; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Consider
a separable Hilbert space H with H ⊆ H continuously and densely,
strongly measurable S : (t1, t2) ∈ [0, T ]2 : t1 < t2 → L(H,H) with
∀ 0 ≤ t1 < t2 < t3 ≤ T : St2,t3 St1,t2 = St1,t3
predictable X : [0, T ]× Ω→ H, Y : [0, T ]× Ω→ H, andZ : [0, T ]× Ω→ HS(U, H) such that for all t ∈ (0, T ] it holds P-a.s. that∫ t
0 ‖Ss,t Ys‖H + ‖Ss,t Zs‖2HS(U,H) ds <∞ and
Xt = S0,t X0 +
∫ t
0Ss,t Ys ds +
∫ t
0Ss,t Zs dWs.
Then X is called a mild Itô process (with evolution family S, mild drift Y , and milddiffusion Z ).
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Theorem (Mild Itô formula; Da Prato, J & Röckner 2012 Trans. AMS (to appear))
Let X be a mild Itô process with evolution family S, mild drift Y , and mild diffusion Z .Then for all ϕ ∈ C2(H, V), t0, t ∈ [0, T ] with t0 < t it holds P-a.s. that∫ t
t0
‖ϕ′(Ss,t Xs)Ss,t Ys‖V + ‖ϕ′(Ss,t Xs)Ss,t Zs‖2HS(U,V) ds <∞,
∫ t
t0
‖ϕ′′(Ss,t Xs)‖L(2)(H,V) ‖Ss,t Zs‖2HS(U,H) ds <∞
and
ϕ(Xt) = ϕ(St0,t Xt0 ) +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Ys ds +
∫ t
t0
ϕ′(Ss,t Xs) Ss,t Zs dWs
+1
2
∑u∈U
∫ t
t0
ϕ′′(Ss,t Xs) (Ss,t Zsu, Ss,t Zsu) ds.
Thanks for your attention!
Thanks for your attention!