brownian motion on manifold - purdue universityfeng71/talk/bm on manifold.pdf · for a di...

Brownian Motion on Manifold

QI FENG

Purdue University

[email protected]

August 31, 2014

QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26

Overview

1 Extrinsic construction of Brownian motionDiffusion ProcessBrownian motion by embedding

2 Intrinsic construction of Brownian motionElls-Elworthy-Malliavin construction of Brownian motionExamples of Brownian motion on Riemannian Manifold


Extrinsic construction of Brownian motion Diffusion Process

SDE on manifolds

Definition

For a differentiable manifold M. An M-valued path x with explosion timee = e(x) > 0 is a continuous map x : [0,∞)→ M̂ such thatxt ∈ M, for 0 ≤ t ≤ e and xt = ∂M for all t ≥ e if e


SDE on manifolds

Definition (semimartingale on manifolds )

Let M be a differentiable manifold and (Ω,F∗,P) a filtered probabilityspace. Let τ be an F∗-stopping time. A continuous, M-valued process Xdefined on[0, τ) is called an M-valued semimartingale if f (X ) is areal-valued semimartingale on [0, τ) for all f ∈ C∞(M).

Definition (SDE on manifolds)

An M-valued semimartingale X defined up to a stopping time τ is asolution of SDE (V1, ...,Vl ,Z ,X0) up to τ if for all f ∈ C∞(M),

f (Xt) = f (X0) +

∫ t0Vαf (Xs) ◦ dZαs

SDE (V1, ...,Vl ,Z ,X0):dXt = Vαf (Xs) ◦ dZαs



Diffusion process

Definition (diffusion process)

(i)An F∗-adapted stochastic process X : Ω→W (M) defined on a filteredprobability space (Ω,F∗,P) is called a diffusion process generated by L if Xis M-valued F∗ semimartingale up to e(X) and

M f (Xt) = f (Xt)− f (X0)−∫ t

0Lf (Xs)ds, 0 ≤ t ≤ e(X )

is a local F∗−martingale for all f ∈ C∞(M).(ii)A probability measure υ on the standard filtered path space(W (M),B(W (M))∗) is called a diffusion measure generated by L if

M f (ω)t = f (ωt)− f (ω0)−∫ t

0Lf (ωs)ds, 0 ≤ t ≤ e(ω),

is a local B(W (M))∗-martingale for all f ∈ C∞(M).QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 5 / 26


Diffusion process

relation between L-diffusion measure and L-diffusion process

X is an L-diffusion⇒ υX = P ◦ X−1 is an L−diffusion meassure.υ is an L-diffusion measure on W (M)⇒ X (ω)t = ωt is an L-diffusionprocess.

local coordinate representation of L

L =1

2

d∑i ,j=1

ai ,j(x)∂

∂x i∂

∂x j+

d∑i=1

bi (x)∂

∂x i,

where a = aij ;U → S+(d), b = bi : U → Rd are smooth functions. U is aneighborhood on M covered by the local coordinate system.

dXt = b(Xt)dt + σ(Xt)dBt ←→ L =∑

bi∂∂xi

+∑

i ,j [σσT ]i ,j

∂2

∂xi∂xj



Properties about diffusion process

Theorem (1.3.4 by Elton P. Hsu)

Let L be a smooth second order elliptic operator on a differentiablemanifold M and µ0 a probability measure on M. Then there exists anL-diffusion measure with initial distribution µ0.

Theorem (1.3.6 by Elton P. Hsu)

An L-diffusion measure with a given initial distribution is unique.


Extrinsic construction of Brownian motion Brownian motion by embedding

Brownian motion on euclidean space

Brownian motion on Rn, transition denssity function:

p(t, x , y) = (1

2πt)n/2e−|x−y |

2/2t

infinitesimal generator:

1

24 = 1

2

n∑i=1

∂2

∂x2i

Note:p(t, x , y) is the smallest positive solution of :

∂p

∂t=

1

24u, lim

t↓0p(t, x , y) = δx(y)



Brownian motion on a Riemannian manifold

Definition (Laplace-Beltrami operator)

The Laplace-Beltrami operator 4M on a Riemannian manifold is:

4M f = div(grad f )

Remark:

〈·, ·〉x = ds2x = gijdx idx j : Riemannian metric on M.〈gradf ,X 〉 = X (f ),X ∈ Γ(M)∫M X (f )dυ = −

∫M fdivXdυ. υ : Riemannian volume measure.

Local coordinates representation

4M f = 1√G∂∂x j

(√Gg ij ∂f

∂x i) = g ij ∂

2f∂x i∂x j

+ bi ∂f∂x i

, bi = 1√G

∂(√Gg ij )∂x j



extrinsic construction of Brownian motion

Stratonovich integral

A general SDE in Stratonovich form: dXt = Vα(X )t ◦ dW αt + V0(Xt)dtIt generates a diffusion process with infinitesimal generator :L = 12

∑li=1 V

2α + V0

Theorem (Whitney’s embedding theorem)

Suppose that M is a differentiable manifold. Then there exists anembedding i : M → RN for some N such that the image i(M) is a closedsubset of RN .(N = 2 dimM + 1 will do)

Theorem (Nash’s embedding theorem)

Every Riemannian manifold can be isometrically embedded in someeuclidean space with the standard metric.




square representation of 4M4M =

∑lα P

2α

Pα :orthogonal projection : ξα → TxM. vector field on M.{ξα} : standard orthonormal basis on Rl ,∀x ∈ M, proof file up later

Brownian motion by embedding

Consider Stratonovich SDE on M driven by a l-dim euclidean Brownianmotion W:

dXt = Pα(Xt) ◦ dW αt , X0 ∈ M.

The solution is a diffusion process generated by 12∑l

α=1 P2α =

124M . Any

M-valued diffusion process generated by 4M/2 is called a Brownianmotion on M.





∑lα P

2α

Pα :orthogonal projection : ξα → TxM. vector field on M.{ξα} : standard orthonormal basis on Rl ,∀x ∈ M,

proof file up later





α=1 P2α =

124M . Any






∑lα P

2α

Pα :orthogonal projection : ξα → TxM. vector field on M.{ξα} : standard orthonormal basis on Rl ,∀x ∈ M, proof file up later





α=1 P2α =

124M . Any



Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion

intrinsic construction of Brownian motion

Idea

W :euclidean Brownian motion → M-valued Brownian motion.

M: Riemannian manifold with Levi-Civita connection ∇ andLaplave-Beltrami operator 4M . Given a probability measureµ,∃!4M/2-diffusion measure Pµ on (W (M),B∗). Any 4M/2-diffusionmeasure on W(M) is called a Wiener measure on W(M).Roughly speaking: Brownian motion on M is any M-valued stochasticprocess X whose law is a Wiener measure on the path space W(M).



differential geometry concepts

connection

x → TxMtangent space → TMtangent bundle → Γ(TM)A connection O : Γ(TM)× Γ(TM)→ Γ(TM) satisfies:1)OfX+gYZ = f OXZ + gOYZ ,, 2)OX (Y + Z ) = OXY + OXZ3) OX (fY ) = f OXY + X (f )YOXY : covariant differentiation to Y along X.

A vector field V along a curve {xt} on M is parallel along the curve ifOẋV = 0 at each point of the curve.

frame bundle

A frame at x is R−linear isomorphism u : Rd → TxM, ei 7−→ ueiThe frame bundle :F (M) = ∪x∈MF (M)xF (M)x : space of all frames at x,



Horizontal lift and stochastic development

cannonical projection π : F (M)→ MA tangent vector X ∈ TuF (M) is called vertical if it is tangent to the fiberF (M)πu. VuF (M) : space of vertical vectors.The curve {ut} is horizontal if each e ∈ Rd , the the vector field {ute} isparallel along {πut}.A tangent vector X ∈ TuF (M) is horizontal if it is the tangent vector of ahorizontal curve form u.We have the relation:

TuF (M) = VuF (M)⊗ HuF (M)

If we consider the orthonormal frames of the tangent space :

TuO(M) = HuO(M)⊗ VuO(M)




The projection π : O(M)→ M induces an isomorphismπ∗ : HuO(M)→ TxM. At each u ∈ O(M) Hi is the unique horizontalvector in HuO(M) whose projection is the ith unit vector uei of theorthonormal frame: π∗Hi (u) = uei , Hi (u) ∈ HuO(M) Hi (u) is thehorizontal lift of uei .(in general, ∀e ∈ Rd)

Bochner’s horizontal Laplacian

4O(M) =∑n

i=1 H2i is called the Bochner’s horizontal Laplacian on O(M)

Important relation

4M f (X ) = 4O(M)(f ◦ π)(u), ∀ u ∈ O(M) with πu = x .



horizontal lift and stochastic development

Let {ut} be a horizontal lift of differentiable curve {xt} on M. sinceẋt ∈ TxtM, so u−1t ẋt ∈ Rd . The anti-development of the curve {xt} is acurve {ωt} in Rd defined by:

ωt =

∫ t0u−1s ẋsds

The anti-development {ωt} and the horizontal lift {ut} of a curve {xt} areconnected by an ODE:

u̇t = Hi (ut)ω̇it




u̇t = Hi (ut)ω̇it ←→ dUt =∑d

i=1 Hi (Ut) ◦ dW it .

Definition

An F(M)-valued semimartingale U is said to be horizontal if thereexists an Rd−valued semimartingale W such that the above ODEholds. The unique W is called the anti-development of U(or of itsprojectionx = πU)

Ler W be an Rd−valued semimartingale and U0 an F(M)-valued,F0−measurable random variable. The solution U of the above SDE iscalled a stochastic development of W in F(M). Its projection X = πUis called a stochastic development of W in M.

Let X be an M-valued semimartingale. An F(M)-valued horizontalsemimartingale such that its projection πU = X is called a horizontallift of X.




W ←→ U ←→ XLemma 2.3.3

Suppose that M is a closed submanifold of RN . For each x ∈ M, letP(x) : RN → TxM be the orthogonal projection form RN to the tangentspace TxM. If X is an M-valued semimartingale, thenXt = X0 +

∫ t0 P(Xs) ◦ dXs .

Theorem 2.3.4

A horizontal semimartingale U on the frame bundle F(M) has a uniqueanti-development W. In fact,Wt =

∫ t0 U−1s Pα(Xs) ◦ dXαs , where Xt = πUt .



horizontal lift and stochastic lift

Theorem 2.3.5

Suppose that X = {Xt , 0 ≤ t ≤ τ} is a semimartingale on M up tostopping tie τ , and U0 an F(M)-valued F0-random variable such thatπU0 = X0. Then there is a unique horizontal lift {Ut , 0 ≤ t ≤ τ} of Xstarting from U0.

Proposition 2.3.8

Let a semimartingale X on a manifold M be the solution ofSDE (V1, · · · ,VN ;Z ,X0) and let V ∗α be the horizontal lift of Vα to theframe bundle F(M). Then the horiaontal lift U of X is the solution ofSDE (V ∗! , · · · ,V ∗N ;Z ,U0), and the anti-development of X is given byWt =

∫ t0 U−1s Vα(Xs) ◦ dZαs .



Equivalent definition of Brownian motion on M

Equivalent definition

Let X : Ω→W (M) be a measurable map defined on a probability space(Ω,F ,P). Let µ = P ◦ X−10 be its initial distribution. Then the followingstatements are equivalent.

X is a 4M/2-diffusion process(a solution to the martingale problemfor 4M/2 with respect to its own filtration FX∗ ), i.e.,M f (X )t := f (Xt)− f (X0)− 12

∫ 10 4M f (Xs)ds, 0 ≤ t ≤ e(X ), is an

FX∗ -local martingale for all f ∈ C∞(M).The law PX = P ◦ X−1 is a Wiener measure on(W (M),B(W (M))), i .e.,PX = Pµ.X is a FX∗ -semimartingale on M whose anti-development is a standardeuclidean Brownian motion.

An M-valued process X satisfying any of the above conditions is called a(Riemannian) Brownian motion on M.



Horizontal Brownian motion

Proposition 3.2.2

Let U : Ω→W (O(M)) be a measurable map defined on a probabilityspace (Ω,F ,P). Then the following statements are equivalent.

U is a 4O(M)/2−diffusion with respect to its own filtration FU∗ .U is a horizontal FU∗ −semimartingale on M whose projection X = πUis a Brownian motion on M.

U is a horizontal FU∗ -semimartingale on M whose anti-development isa standard euclidean Brownian motion.

An O(M)-valued process U satisfying any fo the above condition is calleda horizontal Brownian motion on O(M).


Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold

Brownian motion on a circle

Consider the compact manifold:

S1 = {e iθ : 0 ≤ θ ≤ 2π} ⊂ R2

W: a Brownian motion on R1. ThenThe Brownian motion on S1 is given by Xt = e iWt .The anti-development of X is W.



Brownian motion on a sphere

Consider the d-sphere embedded in Rd + 1.

Sd = {x ∈ Rd+1 : |x |2 = 1}

The projection to the tangent sphere at x is given by

P(x)ξ = ξ − 〈ξ, x〉x , x ∈ Sd , ξ ∈ Rd+1

then the matrix P(x) is P(x)ij = δij − xixj .The Brownian motion on Sd is the solution of the equation :

X it = Xi0 +

∫ t0

(δij − X isX js ) ◦ dW js , X0 ∈ Sd .

This is called the stroock’s representation of spherical Brownian motion.



Brownian motion on a radially symmetric manifold

Consider a radially symmetric manifold.In the polar coordinates(r , θ) ∈ R+ × Sd+1 determined by the exponential map at its pole, themetric has a special form:

ds2 = dr2 + G (r)2dθ2,

where dθ2: Riemannian metric on Sd−1, G is a smooth function on aninterval [0,D) with G (0) = 0, G ′(0) = 1.The Laplace-beltrami operator has the form:

4M = Lr +1

G (r)24Sd−1 .

Where Lr is the radial Laplacian

Lr = (∂

∂r)2 + (d − 1)G

′(r)

G (r)

∂

∂r.




Let Xt = (rt , θt) be a Brownian motion on a radially symmetric manifoldM in polar coordinates.We can construct a Brownian motion on a radially symmetric manifold asa warped product.{rt}: a diffusion process generated by the radial Laplacian Lr{zt}: an independent Brownian motion on ∼d−1.By some rescaling, t → Xt = (rt , zlt ) is a Brownian motion on the radiallysymmetric manifold.

Using martingale property of the Brownian motion X to the distance

function f (r , θ) = r we have rt = r0 + βt +d−1

2

∫ t0

G ′(rs)G(rs)

ds.

For angular process, consider f ∈ C∞(Sd−1). Thenf (θt) = f (θ0) + M

ft +

12

∫ t0

4Sd−1f (θs)G(rs)2

ds,scaling by lt =∫ t

0ds

G(rs)2. We have

f (zt) = f (z0) + Mfτt +

12

∫ t0 4Sd−1f (zs)ds, where zt = θτt , τt inverse of lt .




Let Xt = (rt , θt) be a Brownian motion on a radially symmetric manifoldM in polar coordinates.We can construct a Brownian motion on a radially symmetric manifold asa warped product.{rt}: a diffusion process generated by the radial Laplacian Lr{zt}: an independent Brownian motion on ∼d−1.By some rescaling, t → Xt = (rt , zlt ) is a Brownian motion on the radiallysymmetric manifold.Using martingale property of the Brownian motion X to the distance

function f (r , θ) = r we have rt = r0 + βt +d−1

2

∫ t0

G ′(rs)G(rs)

ds.

For angular process, consider f ∈ C∞(Sd−1). Thenf (θt) = f (θ0) + M

ft +

12

∫ t0

4Sd−1f (θs)G(rs)2

ds,scaling by lt =∫ t

0ds

G(rs)2. We have

f (zt) = f (z0) + Mfτt +

12

∫ t0 4Sd−1f (zs)ds, where zt = θτt , τt inverse of lt .



The EndThis slides is mainly based on firstthree chapters of Elton Hsu’s book”Analysis on Manifolds”.


Extrinsic construction of Brownian motionDiffusion ProcessBrownian motion by embedding

Intrinsic construction of Brownian motionElls-Elworthy-Malliavin construction of Brownian motionExamples of Brownian motion on Riemannian Manifold

brownian motion on manifold - purdue universityfeng71/talk/bm on manifold.pdf · for a di...

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