martingales on manifolds and stochastic riemannian...
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Martingales on manifolds and stochastic
Riemannian geometry ∗
Anton Thalmaier
20th June 2006
Abstract. It is well-known that Brownian motion and martingales on manifolds or
vector bundles connect local and global geometry in an intrinsic way, and that many
questions related to the geometry of Laplace operators have are direct probabilistic
counterpart. It turned out that already the definition of martingales (as driftless
motions with respect to the given geometry) leads to non-trivial questions, since taking
conditional expectations of random variables is by nature a linear operation, ruling
out immediate generalizations to curved spaces.
In this series of lectures we start by introducing some basic concepts of Stochastic
Analysis on manifolds, and proceed then to applications, mainly linear and nonlinear
PDEs, having their origin in Analysis, Geometry and Mathematical Physics.
We finally want to stress the point that the same methods which lead to Harnack
type inequalities in the setting of Riemannian Geometry can be used in Mathematical
Finance to calculate price sensitivities (so-called Greeks) and allow there to stabilize
numerically Monte Carlo approximations.
1 Motivation
Let M be a manifold. We can think of the Brownian motion on M , BM(M),and the set of martingales on M , Mart(M), as the following objects:
BM(M) := continuous limit of geodesic random walkMart(M) := driftless motions
The first definition characterizes semimartingales on M via real-valued semi-martingales.
∗These notes are based on lectures given by Anton Thalmaier during the 28th Finnish
Summer School in Probability Theory, which was held in June 2006. The notes are based
partly on the lecturer’s slides and partly on notes taken by two participants of the Summer
School. Notes contain some material that was not covered during the lectures. The typist
hopes to be informed of any errors found in these notes. Send corrections to the address
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Definition 1.1 Let X be a continuous adapted process taking values in M . X
is a semimartingale ⇔ f ◦X is a real semimartingale for all f ∈ C2(M).
Notation:TM →π M tangent bundle (i.e. TM = ∪x∈MTxM , disjoint union of tangent
spaces attached to x; π(TxM) = {x}).We have
Γ(TM) = {A : M → TM | π ◦A = idM}=: {A : C∞(M) → C∞(M) R− linear | A(fg) = fA(g) + gA(f),∀f, g ∈ C∞(M)}.
(note that π ◦A = idM is equivalent to A(x) ∈ TxM , for all x ∈ M , and for thesecond equality, note that Af(x) = A(x)f = dfx · A(x)), where dfx : TxM → Ris linear.Flow to a vector fieldLet A ∈ Γ(TM). Suppose that φt satisfies
{ddtφt = A(φt)φ0 = idM
i.e. for all f ∈ C∞c (M){
ddt (f ◦ φt) = A(f)(φt)f ◦ φ0 = f
which is equivalent to f ◦ φt(x) − f(x) − ∫ t
0A(f) ◦ φs(x)ds ≡ 0, t ≥ 0, for all
f ∈ C∞c (M).The mapping φ·(x), t 7→ φt(x) is called flow line to A (or integral curve) (startingat x).Note the semigroup nature: Ptf = f ◦ φt implies that d
dt |t=0 Ptf = A(f).Flow to a 2nd order PDOe.g. L = A0 + 1
2
∑ri=1 A2
i , where A0, A1, . . . , Ar ∈ Γ(TM).
Example 1.2 Take M = Rn, A0 = 0, Ai = ∂∂xi
. Then L = 12∆. ¥
Definition 1.3 A continuous adapted process X(x) ≡ (Xt(x))t≥0 with valuesin M is called a flow process or L-diffusion if for all f ∈ C∞c (M) and t ≥ 0
Nft (x) := f ◦Xt(x)− f(x)−
∫ t
0
Lf(Xs(x))ds (1)
is a martingale, i.e.
EFs
[f ◦Xt(x)− f ◦Xs(x)−
∫ t
s
Lf(Xr(x))dr
]= 0, ∀s ≤ t.
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Note that for Nft (x) we have
E[Nf
t (x)]
= E[Nf
0 (x)]
= 0
⇒ Ptf(x) = E [f ◦Xt(x)] = f(x) +∫ t
0E [Lf(Xs(x))] ds
⇒ ddt |t=0 Ptf(x) = Lf(x).
Remark 1.4 Suppose that Un ↑ M are open and relatively compact sets suchthat Un ⊂ Un ⊂ Un+1 ⊂ . . . and ∪nUn = M . Define
τn = inf{t ≥ 0 | Xt(x) 6∈ Un} ↑ supn
τn =: ξ(x),
with ξ(x) being the lifetime of X·(x). Then
{ξ(x) < ∞} ⊂{
limt↑ξ(x)
Xt(x) = ∞ in M = M ∪ {∞}}
,
almost surely.
Remark 1.5 For all f ∈ C∞c (M), the process Nft (x) is a local martingale on
[0, ξ(x)).
Remark 1.6 Basically we have the following equivalence: f(X·(x)) has “nicepaths“ for all f ∈ C∞c (M), x ∈ M ⇔ L is a vector field.
Definition 1.7 Let M be a Riemannian manifold and L = 12∆M the Laplacian
on M . Then {L-diffusions} := {B | B is a BM(M)}.
1.1 L-diffusions: why useful?
1.1.1 Dirichlet problem
In the Dirichlet problem we are given an open, connected and relatively compactnonempty set D ⊂ M and a function φ ∈ C(∂D). We wish to find a functionu ∈ C∞(D) ∩ C(D) such that
{Lu = 0 on D
u |∂D= φ.
Suppose there is an L-diffusion X. Let the open sets Dn ⊂ D be such thatDn ↑ D. Define
τn := inf{t ≥ 0 | Xt(x) 6∈ Dn} ↑ τ(x) = inf{t ≥ 0 | Xt(x) 6∈ D},
and let un ∈ C∞c (M) be such that un | Dn = u | Dn and supp(un) ⊂ D. Then
Nt(x) := un ◦Xt(x)− un(x)−∫ tτn
0
(Lun)Xr(x)dr
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is a martingale, which in turn implies that
Ntτn(x) = un ◦Xtτn(x)− un(x)− ∫ t
0(Lun)Xr(x)dr
u ◦Xtτn− u(x)− ∫ tτn
00dr
is a martingale, and since E[Ntτn(x)] = 0, we have
u(x) = E [u ◦Xtτn(x)]
= limn→∞ E [u ◦Xtτn(x)]
= E[u ◦Xtτ(x)(x)
]
= E [u ◦Xτn(x)] = E [φ ◦Xtτn
(x)] ,
where we used τn ↑ τ(x) to obtain the third equality and assumed τ(x) < ∞a.s. to obtain the fourth equality.
Now we get
u(x) =∫
∂D
φdµx,
where µx(A) := P{Xτ(x)(x) ∈ A} for any measurable A ⊂ ∂D.Moral
(a) hypothesis τ(x) < ∞ a.s. for all x ∈ D implies uniqueness for theDirichlet problem.
(b) hypothesis τ(x) → 0 in probability, if D 3 x → a ∈ ∂D implies that
u(x) = E[φ ◦Xτ(x)(x)] → φ(a)
if D 3 x → a ∈ ∂D.
Example 1.8 Let M = R2 \ {0}, L = ∂2
∂ν2 and D = {x ∈ R2 | r1 <| x |< r2},where 0 < r1 < r2. Now, if u is a solution of the Dirichlet problem, then u+v(r)is a solution as well, where the radial function v is such that v(r1) = v(r2) = 0.¥
Example 1.9 Let M = R2, L = ∂2
∂x21
and suppose that D is as in the picture.Then there exists a solution to the Dirichlet problem, if and only if φ(a) =(1/2)[φ(b) + φ(c)]. ¥
1.1.2 Heat equation
The heat equation is given by{
∂∂tu = Lu
u |t=0= f.
Suppose X·(x) is an L-diffusion. Then (t,Xt(x)) is an L-diffusion (where L =∂∂t + L).
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Hypothesis: ξ(x) < ∞ a.s. for all x ∈ M . (∗)Let u be a bounded solution of the heat equation. Fix t > 0 and suppose
0 ≤ s ≤ t. Then
u(t− s,Xs(x))− u(t, x)−∫ s
0
(∂
∂r+ L
)u(t− r,Xr(x))dr
is a local martingale and consequently a martingale, with(
∂∂r + L
)u(t−r,Xr(x)) =
0. Thus
u(t, x) = E [u(t− s,Xs(x))] →s↑t E [u(0, Xt(x))] = E [f ◦Xt(x)] .
MoralUnder hypothesis (∗), we have uniqueness of bounded solutions of the heat
equation.
2 Construction of flow processes (L-diffusions)
e.g. L = A0 + 12
∑ri=1 A2
i .
2.1 SDEs on M
(1) Suppose E is a finite-dimensional vector space, dim(M) = r. Then we havethe following diagram:
A
M × E →A TM
pr1 ↓ id ↓ π
M → M,
where (x, e) 7→ A(x)e is such that A(x) : E → TxM is linear for all x ∈ M .(2) Suppose Z is an E-valued semimartingale and denote the Stratonovich
differential by ∗. We have
dX = A(x) ∗ dZ or dX =r∑
i=1
Ai(X) ∗ dZi,
where Ai = A(·)ei ∈ Γ(TM) and (e1, . . . , er) is the basis of E.
Example 2.1 Let E = Rr+1, Z = (t, W 1, W 2, . . . , W r) with (W 1, . . . ,W r)being a BM(Rr) and (e0, e1, . . . , er) the canonical basis of Rr+1. Now
dX = A0(X)dt +r∑
i=1
Ai(X) ∗ dZi,
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i.e. for all f ∈ C∞c (M)
d(f ◦X) = A0(f)(X)dt +r∑
i=1
(Aif)(X) ∗ dZi, (2)
which implies that
d(f ◦X)− Lf(X)dt = d(martingale),
i.e. maximal solutions of 2 are L-diffusions. ¥
Example 2.2 (Stroock’s construction of BM(M)) Let M be a Riemannianmanifold and M ↪→ Rl Whitney embedding. Suppose A(x) : Rl → TxM isan orthogonal projection and define X by
dX = A(X) ∗ dW =l∑
i=1
Ai(X) ∗ dW i, ,
where Ai := A(·)ei ∈ Γ(TM). Then X is BM(M). ¥
3 Quadratic variation and integration of 1-forms
Let X be a semimartingale taking values in M . Suppose that α ∈ Γ(T ∗M),i.e. αx : TxM → R is linear for all x ∈ M , and that b ∈ Γ(T ∗M ⊗ T ∗M), i.e.bx : TxM × TxM → R is bilinear for all x ∈ M .Goal: to explain
∫X
α and∫
b(dX, dX).
Lemma 3.1 Let M be a manifold. Then there exist h1, . . . , hl ∈ C∞(M) suchthat
(i) all f ∈ C∞(M) can be written as f = f ◦ (h1, . . . , hl) =: f ◦ h, i.e.
→f RM ↑ f
→h Rl
with f ∈ C∞(Rl,R);
(ii) all α ∈ Γ(T ∗M) can be written as α =∑l
i=1 αidhi, where αi ∈ C∞(M);
(iii) all b ∈ Γ(T ∗M ⊗ T ∗M) can be written as b =∑l
ij=1 bijdhi ⊗ dhj, wherebi ∈ C∞(M).
Idea: Take M ↪→h Rl to be Whitney embedding. Then h(M) ⊂ Rl is a closedsubmanifold.
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We denote the vector space of continuous real semimartingales by
S := M⊕A,
where M is the space of local martingales and A is the space of processes oflocally bounded variation such that A0 = 0.
Proposition 3.2 Let X be a continuous semimartingale. Then there exists aunique linear map from Γ(T ∗M) to S such that α 7→ ∫
Xα =
∫α(∗dX) =: IX(α)
and for all f ∈ C∞(M)
(i) df 7→ f ◦X − f(X0) and
(ii) f · α 7→ ∫(f ◦X) ∗ dIX(α).
We call∫
Xα the Stratonovich integral of α along X.
Proof. Since α ∈ Γ(T ∗M), we have the representation α =∑
i αidhi and so∫
X
α =∑
i
∫(αi ◦X) ∗ d(hi ◦X). (3)
Uniqueness is now clear. It remains to show that 3 is well-defined, that is, thatwe have the implication
α =∑
finite
uνdfν = 0 ⇒∑
ν
(uν ◦X) ∗ d(fν ◦X) = 0.
Corollary 3.3 (pullback) Suppose that α ∈ Γ(T ∗N) and φ : M → N belongsto C∞. Then φ∗α ∈ Γ(T ∗M) and (φ∗α)x(ν) = αφ(x)(dφx ◦ ν). Furthermore, if X
is a semimartingale taking values in M , then∫
X
φ∗α =∫
φ◦Xα
Proof. This is merely checking that∫
Xφ∗α satisfies the defining properties of∫
φ◦X α, likeφ∗(df) = d(f ◦ φ).
Corollary 3.4 Suppose that Xt = x(t) is a C1 curve in M . Then∫
X
α =∫
(αx(t))dt.
Proof. Again, just check the defining properties, e.g. α = df
∫df(x(t))dt =
∫˙(f ◦ x)(t)dt = f ◦ x(·)− f(x(0)).
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Proposition 3.5 Let X be as in Proposition 1 and let b be bilinear. Then thereis a unique linear map from Γ(T ∗M ⊗ T ∗) to S such that b 7→ ∫
b(dX, dX) =:IX(b) and
(i) df ⊗ dg 7→ [f ◦X, g ◦X] and
(ii) f · b 7→ ∫(f ◦X)b(dX, dX),
for all f, g ∈ C∞(M).∫
b(dX, dX) is “b-quadratic variation“ of X; write(∫
b(dX, dX))t=
∫ t
0b(dX, dX).
Idea: b ∈ Γ(T ∗M ⊗ T ∗) implies that b =∑
finite bijdhi ⊗ dhj . Thus∫
b(dX, dX) :=∑
ij
∫(bij ◦X) d[hi ◦X,hj ◦X].
To show: well-defined!
Remark 3.6∫
b(dX, dX) depends only on the symmetric part of b.
Remark 3.7 (pullback) If b ∈ Γ(T ∗M ⊗ T ∗) and φ : M → N is C∞, then∫
φ∗b(dX, dX) =∫
b(dφ ◦X, dφ ◦X).
4 Connections and martingales
We need additional structure: connection in E = TM . There are 3 ways ofintroducing a connection in a vector bundle E.
• covariant derivation in E;
• parallel transport //t : Eα(0) →∼ Eα(t) in E;
• horizontal splitting.
Definition 4.1 Let E →π M be a vector bundle. A connection (covariantderivative) in E is a map
∇ : Γ(TM)× Γ(E) → Γ(E)(A,B) 7→ ∇AB
such that
∇ is C∞-linear in the first argument:
∇A1+A2B = ∇A1B +∇A2B
∇fAB = f · ∇AB, f ∈ C∞(M)
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∇ is a derivation in the second argument:
∇A(B1 + B2) = ∇AB1 +∇AB2
∇A(fB) = A(f)B + f · ∇AB, f ∈ C∞(M).
Horizontal splittingLet e ∈ E. Define
Ve = {v ∈ TeE | (dπ)ev = 0} = ker(dπ)e = TeEπ(e) = TEx ' Ex,
where ' denotes isomorphism, and
He = {v ∈ TeE | v horizontal},
i.e. v is velocity of a parallel curve starting from e. Informally, Ex = Ve are“vertical vectors“ and He are “horizontal vectors“. We have
TeE = Ve ⊕He, i.e. TE = π∗E ⊕H.
Remark 4.2 Let E = TM and γ a curve in M . Then γ is geodesic ⇔ γ isparallel along γ (⇔ ∇γ γ = 0).
For every connection ∇ in E we have an induced connection in E?,
∇ : Γ(TM)× Γ(E∗) → Γ(E∗)(A, b) 7→ ∇Ab
Let B ∈ Γ(E). Then
A(bB) = (∇Ab)(B) + b(∇AB),
and we can take this as definition of (∇Ab)(B)!
Example 4.3 Suppose E = TM and f ∈ C∞(M). Then for df ∈ Γ(T ∗M),∇df ∈ Γ(T ∗M ⊗ T ∗M) and we can define the Hessian of f as follows:
∇df(A,B) := (∇Adf)(B). ¥
Definition 4.4 Let∇ be a connection on M (in TM) and let X be a continuoussemimartingale taking values in M . Then X is a ∇-martingale, if and only iffor all f ∈ C∞(M)
d(f ◦X)− 12∇df(dX, dX) = 0.
Compatibility with a metricLet (M, g) be a Riemannian manifold and g ∈ Γ(T ∗M ⊗ T ∗M) such that
gx =<,>x: TxM × TxM → R
is symmetric and positive definite for all x ∈ M .
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Definition 4.5 Let ∇ be a connection on M . ∇ is a metric connection, if andonly if all parallel transports //t : Tα(0)M → Tα(t)M are isometries.
Lemma 4.6 (Levi–Civita) On a Riemannian manifold (M, g) there exists aunique metric connection such that (A,B) 7→ ∇df(A,B) is symmetric for allf ∈ C∞(M). We call this the Levi–Civita (LC) connection. If f ∈ C∞(M),then we have the following relations:
∇df ∈ Γ(T ∗M ⊗ T ∗M)∆Mf := trace∇df ∈ C∞(M)∆Mf(x) =
∑ni=1∇df(ei, ei),
where (e1, . . . , en) is an orthonormal basis of TxM .
Definition 4.7 Let X be a semimartingale with values in (M, g). Then X is aBrownian motion on M , BM(M), if and only if
d(f ◦X)− 12∆Mf(X)dt = 0,
for all f ∈ C∞(M).
Lemma 4.8 (Levy characterization of BM) Let (M, g) be a Riemannianmanifold and ∇ the LC connection. For an M -valued semimartingale the fol-lowing are equivalent:
1. X is BM(M);
2. X is a ∇-martingale and∫
b(dX, dX) =∫
(trace b) (X)dt,
for all b ∈ Γ(T ∗M ⊗ T ∗M);
3. X is a ∇-martingale and
[f ◦X, f ◦X] =∫‖ grad f ‖2(X)dt,
for all f ∈ C∞(M).
5 Moving frames
Let M be a Riemannian manifold. Define
O(M) = ∪x∈MOx(M),
10
where Ox(M) = {u : Rn → TxM | u isometry} and
u ∈ Ox(M) := (u1, . . . , un) := (ue1, . . . , uen)
is an orthonormal basis of TxM . We call
O(M) →π M
orthonormal base bundle.Let t 7→ u(t) ∈ P := O(M) be C∞-curve and X = π ◦ u.t 7→ u(t) is horizontal ⇔ t 7→ u(t)ei parallel along t 7→ x(t) (i = 1, . . . , n).If
u ∈ P : Vu = ker(dπ)u ⊂ TuP
Hu = {X ∈ TuP | X horizontal},(horizontal meaning that X = u(0), where u(·) is a horizontal curve in P withu(0) = u), then
TuP = Vu ⊕Hu
= ker(dπ)u ⊕Hu
(dπ)u : Hu →∼ Tπ(u)M
Hu ←hu Tπ(u)M
h : π∗TM →∼ H ⊂ TP
ιu : Vu →∼ g,
where g := Lie algebra to G = O(n) and hu is horizontal lift.Suppose X ∈ Γ(TP ). Then
X = vert X+ hor X
Xu = (vert X)u+ (hor X)u
∈ TuP ∈ Vu ∈ Hu
For L1, . . . , Ln ∈ Γ(TP ), we define the standard horizontal vector fields
Li(u) := hu(uei).
Remark 5.1 L1, . . . , Ln generate H.
Definition 5.2 For ω ∈ Γ(T ∗P, g) , we define the connect form by
ωu(Xu) = ι−1u (vert X)u
and for θ ∈ Γ(T ∗P,Rn) the canonical 1-form by
θu(Xu) = u−1(dπ)uXu,
where (dπ)uXu ∈ Tπ(u)M .
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Definition 5.3 Let X be an M -valued semimartingale. A semimartingale U ,taking values in P = O(M) is called horizontal lift of X, if
1. π ◦ U = X a.s.;
2. U is “horizontal“, i.e.∫
Uω = 0.
Theorem 5.4 Let X be a semimartingale with values in M , X0 = x0, and u0
be an F0-measurable random variable such that π ◦ u0 = x0. Then there existsa unique horizontal lift U of X such that U0 = u0.
Definition 5.5 (Parallel transport along a semimartingale X) Consider
//t
TX0M → TXtM
U−10 ↓ ∼ ↑ Ut
Rn .
We define the parallel transport //t := UtU−10 .
SupposeX 7→ U 7→ Z =
∫U
θ
M O(M) Rn.
Then
dU =n∑
i=1
Li(U) ∗ dZi.
In particular,
d(f ◦ π ◦ U) =n∑
i=1
(df)X(dπ)ULi(U) ∗ dZi,
where (dπ)ULi(U) = Uei.
dX =∑
i
Uei ∗ dZi = U ∗ dZ = //t ∗ dA(X),
where A(X) = U0Z (taking values in TX0M) is the anti-development of X.
dX = //t ∗ dA(X)t.
Geometric Ito formulaFor all f ∈ C∞(M)
d(f ◦X) = df(U ∗ dZ)d(f ◦X) = df(UdZ) + 1
2∇df(dX, dX)
Hence, X is a ∇-martingale ⇔ Z is a local martingale.Classical caseLet M be a Riemannian manifold with the LC connection. The following areequivalent.
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1. Z is a BM(Rn) with generator L = 12∆;
2. U is a horizontal BM on O(M) with generator 12∆hor := 1
2
∑ni=1 L2
i ;
3. X is a BM(M) with generator 12∆M .
In the last equivalence we used that for all f ∈ C∞(M)
∆hor(f ◦ π) = (∆Mf) ◦ π.
CurvatureLet M be a Riemannian manifold with dimM ≥ 2 and let ∇ be the LC connec-tion. We define three curvatures:
1. Riemannian curvature tensor R. Define the C∞(M)-trilinear map
R : Γ(TM)3 → Γ(TM)
by
R(X,Y, Z) ≡ R(X, Y )Z := ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z.
2. Riemannian sectional curvature Sectx(σ). Let u, v ∈ TxM and let
σ = span(u, v) ⊂ TxM
be a plane. Define
Sectx(σ) :=< R(u, v)v, u >
‖ min(u, v) ‖2 ,
e.g. K1 ≤ Sect ≤ K2 means
K1 ≤ Sectx(σ) ≤ K2 ∀σ ⊂ TxM, x ∈ M.
3. Ricci curvature Ric ∈ Γ(T ∗M ⊗ T ∗M) is defined for u, v ∈ TxM by
Ricx(u, v) := trace (w 7→ R(w, u, v)),
where (w 7→ R(w, u, v)) ∈ Hom(TxM, TxM), i.e. Ricx : TxM × TxM → Ris a symmetric bilinear form given by
< Ric]x(u, ·), v >:= Ricx(u, v),
where Ric]x(u, ·) ∈ TxM .
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6 Application to calculation of price sensitivities
Call option (Q is the risk-neutral measure):
VT = VT (x) = (XT (x)−K)+ = φ(XT (x))V0 = V0(x) = EQ[(XT (x)−K)+] = EQ[φ(XT (x))].
Price sensitivityGreek Delta
∆ =∂
∂xV0(x).
Why important? We have Vt(x) = v(t,Xt(x)), where v(t, y) = EQ[φ(XT−t(y))].Then {
VT (x) = (XT (x)−K)+dVt(x) = δtdXt(x)
(perfect hedging), where δt = ∂∂y v(t, y) |y=Xt(x) is the strategy (P.-L. Lions et
al. 1999; 2001).More generally, let X = (Xt(x))t≥0 be a solution to
dX = b(X)dt + σ(X)dW (SDE on Rn)
with b n× 1-dimensional, σ n× r-dimensional and dW r × 1-dimensional.
(1) uT (x) = PT φ(x) = E[φ(Xmin(T,τ)(x))]
(2) (PT φ)(x) = E[φ(XT (x))χ{T<τ}]
(3) u(x) = Pφ(x) = E[φ(Xτ (x))]
τ = inf{t ≥ 0 | Xt(x) 6∈ D}, D ⊂ Rn domain.Note: uT , resp.u, are solutions to certain PDEs, e.g. in case (1)
∂∂T uT = Lu
uT |T=0= φ
uT |∂D= φ |∂D,
where L =∑n
i=1 biDi + 12
∑ni,j=1(σσ∗)ijDiDj , Di = ∂
∂xi.
Φ = φ(Xmin(T,τ)(x)), resp. Φ = φ(XT (x))χ{T<τ}, resp. Φ = φ(Xτ (x)). Then
Mt(x) = EFt [Φ] = uT−t(Xt(x))
(resp. u(Xt(x))) is a family of local martingales depending on a parameter x.This implies that ∂
∂xMt(x) is a local martingale as well.In particular, for all x ∈ Rn
(duT−t)Xt(x)Xt∗ · v, t < min(T, τ),
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is a local martingale (where Xt∗ is differential flow with respect to x). Thus,for lt pathwise absolutely continuous,
ηt := (duT−t)Xt(x)Xt∗ lt −∫ t
0
(duT−r)Xr(x)Xr∗ lrdr
is also a local martingale. But uT−t(Xt(x)) =∫ t
0(duT−r)σ(Xr(x))dWr, hence
mt = (duT−t)Xt(x)Xt∗ lt − uT−t(Xt(x))∫ t
0
< σ−1(Xr(x))Xr∗ lr, dWr >
is a local martingale (where σ−1 is the right-inverse of σ).Choose lt such that
(mt) is a true martingale,
l0 = v and lmin(t,τ) = 0.
Then (duT )xv = E[m0] = E[mmin(T,τ)].
Theorem 6.1 If
Ptφ(x) = E[φ(Xmin(T,τ)(x))] = E[Φ(x)],
then
d(Ptφ)xv = −E[Φ(x)∫ min(T,τ)
0
< σ−1(Xr(x))Xr∗ lr, dWr >],
where l0 = v, lmin(T,τ) = 0.
Relation to geometryLet M = Rn and σ(x) : Rr → Rn = TxRn surjective. Then there existsRiemannian metric g such that σ(x)σ∗(x) = idTxM . Then L = 1
2∆g + V forV ∈ Γ(TM).Filtering out noise (Elworthy–Yor)Let Fr(x) = σ{Xs(x) | s ≤ r} and define Qr(v) := //rEFr(x)//−1
t Xr∗v. Then{
ddr//−n
t Qr(v) = −n2 //−n
r Ric](Qr(v), ·) + //−1r ∇Qr(v)v
Q0(v) = v.
Then
d(Ptφ)xv = −E[Φ(x)∫ min(T,τ)
0
< Qr lr, σ(Xr(x))dWr >]
and (duT−r)Xr(x)Qr · v is a local martingale.e.g. τ = ∞, lr = T−r
T v ⇒ lr = − vT .
Let Qr = Qr · v.
ddr‖ //−n
r Qr ‖2TxM = 2 < ddr //−1
r Qr, //−1r Qr >
= 2 < //−1r [− 1
2Ric](Qr, ·) +∇Qrv], //−1r Qr >
= −Ric(Qr, Qr) + 2∇v(Qr, Qr), .
where ∇v(w, w) :=< ∇wv, w >.
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7 Brownian motion, harmonic maps and Yang–
Mills fields
Geometric variational problems from a probabilistic point of viewe.g. harmonic objects
linear nonlinear
maps harmonic functions M → R harmonic mappings M → N
forms harmonic forms – Hodge theory harmonic curvature – Yang–Mills theory
harmonic↔ variational problem↔ Euler Lagrange equation↔ BM/martingalesReferences
Probab. Theory Relat. Fields 1996
J. Math. Pures Appl. 1998, 2002, 2004
J. Funct. Anal. 1998, 2001
Ann. Inst. H. Poincare 1999
Seminaire de Probabilites 1998, 2003
Annals of Probability 2003
Potential Analysis 2004
Bull. Sci. Math. 2006
7.1 The heat flow for harmonic maps
Let (M, g) and (N,h) be Riemannian manifolds and f ∈ C∞(M, N). Thendf ∈ Γ(T ∗M ⊗ f∗TN), i.e. (df)x : TxM → Tf(x)N is linear.Variational problem(e.g. M compact)Find f (in a given homotopy class) such that
E(f) =∫
M
| df |2d vol = minimum.
Euler–Lagrange: ∆f = 0 (f harmonic)
∇df ∈ Γ(T ∗M ⊗ T ∗M ⊗ f∗TN)(∇df)x : TxM × TxM → Tf(x)N bilinear, symmetric∆f = trace∇df ∈ Γ(f∗TN) tension(∆f)(x) ∈ Tf(x)N
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In coordinates (quasi-linear operator)
(∆f)k = ∆Mfk + (NΓk ◦ f)(df, df) = 0,
with k = 1, . . . , dimN .Time dependent (or parabolic) problem:Harmonic map heat equation
{∂∂tu = 1
2∆u
u |t=0= f(=: u0).
Then E(u(t, ·)) ↓.Deformation to harmonic maps
Remark 7.1 (M , N compact, u0 ∈ C∞(M, N)) If SectN ≤ 0, then the heatflow lives forever:
f = u0 ' u∞ = limt→∞
u(t, ·) ∈ C∞(M, N)
with u∞ harmonic (Eells–Sampson 1964).
In general the harmonic map heat equation may develop singularities (“blow upin finite time“), i.e. there exists T > 0 and x0 ∈ M such that
lim supt↑T
| du(t, ·) |2(x0) = ∞.
Theorem 7.2 (Probab. Theory Relat. Fields 1996) Let T > 0. There existsε = ε(M, N, T ) > 0 such that for each initial condition u0 : M → N in anontrivial homotopy class and of energy E(u0) < ε, the solution of the heatequation
∂
∂tu =
12∆u, u |t=0= u0,
explodes before T .
7.2 Stochastic description
Theorem 7.3 For u : M → N continuous, the following are equivalent:
a) u is harmonic;
b) u maps BM(M, g) to Mart(N,∇).
In other words, if X = (Xt(x))t≥0 is a BM on (M, g), then Y := u(X) is amartingale on (N, h). In addition,
∫ t
0
h(dY, dY ) =∫ t
0
| du |2(Xs)ds.
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Example 7.4 Let u : Rn \ {0} → Sn−1 be the harmonic mapping x 7→ x‖x‖ .
Then (u ◦ Xt(x))t≥0 is a martingale on Sn−1 starting from x‖x‖ , x 6= 0, and
(u ◦Xt(0))t>0 is a martingale without starting point, i.e. Y = u ◦X(0). Then
limε→0
∫ t
ε
h(dY, dY ) = limε→0
∫ t
ε
| du |2(Xs(0))ds = +∞. ¥
Let X be a general process taking values in a Riemannian manifold M . Definethe damped parallel transport Θt : TX0M → TXt
M by the following covariantequation: {
d//−1t Θt = − 1
2//−1t RM (Θt, dXt)dXt
Θ0 = idTx0M,
where //t is the usual parallel transport along X with respect to the Levi–Civitaconnection.
Example 7.5 Let X be BM on (M, g). Then
d//−1t Θt = −1
2//−1
t RicM (Θt)dt.
Consider the following situation:
u : M → N continuous,
X BM on M with X0 = x,
Y := u ◦X on N (Y0 = u(x)),
and let ΘMt : TxM → TXtM , ΘN
t : TY0N → TYtN be the associated dampedparallel transports.Define
• B a BM with values in TxM by dBt = //−1t dXt
• ˜u(X) with values in Tu(x)N by d ˜u(X)t = (ΘNt )−1d(u ◦X)t. ¥
7.3 Nonlinear derivative formulas
Theorem 7.6 (Stochastic representation of the gradient; Arnaudon–Thalmeier,JMPA 1998) Let D 6⊂ M be a relatively compact domain, and let u : D → N bea harmonic map. Then, for each v ∈ TxM , x ∈ D,
(du)xv = −E[
˜u(X)τ
∫ τ
0
< ΘMs ls, //sdBs >
],
where
• τ is a stopping time; τ ≤ inf{t > 0 | Xt(x) 6∈ D}
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• lt is an adapted process with values in TxM such that
(∫ τ
0
| lt |2dt)1/2 ∈ L1+ε, for some ε > 0
and l0 = v, lτ = 0.
Corollary 7.7 (Linear case) If N = R, then ˜u(X)τ = u(Xτ (x)) and one ob-tains a version of a classical formula of Bismut (1984).
7.4 A priori estimates
Theorem 7.8 (Derivative estimates) We have the inequality
| (du)xv |≤‖∫ τ
0
< ΘMs ls, //M
s dBs >‖p · ‖ ˜u(X)τ ‖q, (4)
where 1 ≤ p < ∞ and 1/p + 1/q = 1.
We denote the two factors on the right hand side by (I) and (II).To estimate (I): let p = 2 and | v |≤ 1. Then
(I) ≤√
C(dist(x, ∂D))
where C(r) = π2
4 (n + 3)r−2 + π2
√α(n− 1)r−1 + α with α = −min(k, 0) and k
is a lower bound for Ricci on M .In particular, if
RicM ≥ −(n− 1)K2, K ≥ 0,
then(I) ≤ c(n)
[K +
1r(x)
], r(x) := dist(x, ∂D).
[A. Thalmaier and F.-Y. Wang J. Funct. Anal. (1998)]To estimate (II):
Theorem 7.9 (M. Arnaudon, X.-M. Li, A. Thalmaier, Ann. Inst. H. Poincare(1999)) Let −κ1 ≤ SectN ≤ κ2 with κ1, κ2 ≥ 0. Then
exp(− κ1
2
∫ t
0
h(dY, dY ))≤| (ΘN
t )−1 |≤ exp(κ2
2
∫ t
0
h(dY, dY ))
where h denotes the metric on N and Y := u ◦X.
The linear case: ˜u(X) = u(X(x)). Then for all 1 ≤ p < ∞ we have
| (du)xv |≤‖ u ‖D · ‖∫ τ
0
< ΘMs ls, //M
s dBs >‖p,
where the second factor on the right hand side is (I).
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7.5 Liouville theorems
The stochastic method allows to establish Liouville theorems in different situa-tions (A. Thalmaier and F.-Y. Wang):
• Harmonic maps of small image
• Harmonic maps of sublinear growth
• Non-positively curved targets
• Harmonic maps of bounded dilatation
• Harmonic maps defined on Liouville manifolds
• Harmonic maps of certain asymptotic behaviour
Idea: Establish precise estimates, e.g., on geodesic balls, and exhaust the man-ifold by a sequence of such balls.Harmonic maps of bounded dilatation
Definition 7.10 Let λ1(x) ≥ λ2(x) ≥ . . . ≥ λn(x) ≥ 0 be the eigenvalues of(du)∗x(du)x : TxM → TxM . The map u : M → N is said to be of K-boundeddilatation (for some K > 0) if and only if λ1 ≤ K2(λ2 + . . . + λn) on M .
Theorem 7.11 (non-linear a priori estimate; JMPA 1998) Let M be compactwith ∂M 6= ∅, RicM ≥ k ∈ R, SectN ≤ −β < 0. Let u : M → N be a harmonicmap of K-bounded dilatation. Then
| (du)x |2 ≤ K2
βC (dist(x, ∂M)) , x ∈ M \ ∂M,
where
C(r) =π2
4(n + 3)r−2 +
π
2
√α(n− 1)r−1 + α
and α = −min(k, 0).
Corollary 7.12 Let (M, g), (N, h) be two Riemannian manifolds, M complete,and
RicM ≥ −α
SectN ≤ −βα ≥ 0, β ≥ 0.
If u : M → N is a harmonic map of K-bounded dilatation for some K > 0,then
| (du) |2 ≤ αK2
β.
In particular, if α = 0, then u is constant. (Shen, Crelle 1984).
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7.6 Yang–Mills theory
Let E →π M be a Riemannian vector bundle over M ,
∇ : Γ(E) → Γ(T ∗M ⊗ E)
a metric connection on E, and
A0(E) →d∇ A1(E) →d∇ A2(E) →d∇ . . . →d∇ An(E),
where Ap(E) := Γ(∧pT ∗M ⊗ E) (E-valued differential forms).We have no longer d∇ ◦ d∇ = 0:
d∇ ◦ d∇ = R∇ on A0(E) ≡ Γ(E),
where R∇ ∈ A2(End(E)) is the curvature of ∇.Variational problem
Y M(∇) =∫
M
| R∇ |2 dvol = min
Euler–Lagrange:
(d∇)∗R∇ = 0 (Yang–Mills equations)
Note. Because of d∇R∇ = 0 (Bianchi identity), the following conditions areequivalent:
1. ∇ is Yang–Mills, i.e. (d∇)∗R∇ = 0;
2. ∆(R∇) = 0, where ∆ = d∇(d∇)∗ + (d∇)∗d∇ (harmonic curvature)
7.6.1 Reduction to Stochastic Analysis
Perturbation of the BM by a vector field along X → Variation of the paralleltransport in E.For u ∈ TX0M let
Xt(a, u) = expXt(a//tu), a ∈ (−ε, ε),
and let Jt(a, u) = //a,ut denote the parallel transport in E along t 7→ Xt(a, u).
Consider∇a |a=0 Jt(a, u) ∈ Hom(EX0 , EXt)//−1
t ∇a |a=0 Jt(a, u) ∈ End(EX0)//−1
t ∇a |a=0 Jt(a, ·) ∈ T ∗X0M ⊗ End(EX0).
Note //−1∇J := //−1· ∇a |a=0 J(a, ·).
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Theorem 7.13 (Sem. Prob. 2003, Ann. Prob. 2003) The following condi-tions are equivalent:
(a) ∇ is a Yang–Mills connection;
(b) //−1∇J is a martingale.
The quadratic variation S of //−1∇J is the integral of the Yang–Mills energyalong the paths of the BM X:
St =∫ t
0
| R∇ |2(Xs)ds.
Alternative descriptionRandom holonomy along Brownian bridgesLet x′ := expx(au), u ∈ TxM and Xt(a) a Brownian bridge from x′ to x oflifetime a2 (rescaled to lifetime 1). Denote by τ0,a the parallel transport in E
along the geodesic from x to x′ and by J = J(a) the parallel transport in E
along X(a).
Theorem 7.14 We have
E[J1(a)τ0,a] = idEx −a3
6(d∇)∗R∇(u) + O(a4),
where J1(a)τ0,a ∈ End(Ex). In particular,
∇ is Yang–Mills ⇔ E[J1(a)τ0,a]− idEx = O(a4)
for all x, u as above.
7.7 The parabolic Yang–Mills equation
The parabolic Yang–Mills equation is{
∂∂t∇t = − 1
2 (d∇t
)∗R∇t
∇t |t=0= ∇0.
In general, singularities may develop, i.e., for T > 0, x0 ∈ M ,
lim supt↑T
| R∇t |2(x0) = ∞.
Theorem 7.15 (Non-explosion in dimension 3) If n ≤ 3, then for any initialvalue, the solutions to the Yang–Mills heat equation are defined for all positivetimes.
Theorem 7.16 (Explosion in dimension > 4) Let n > 4 and let E be a non-trivial vector bundle over Sn. Then there exists ε = ε(E) > 0 such that, for anyinitial condition ∇0 with energy Y M(∇0) < ε, explosion appears in finite time.
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Method of proof. (J. Math. Pures Appl. 2002) Any solution (∇s) to the par-abolic Yang–Mills equation on [0, T ) ×M induces canonically a connection ∇on π : E → [0, T )×M , where E is the vector bundle with fibre Es,x := Ex.Let J0,s(a, u) : EX0(a,u) → EXs(a,u), s ≤ t ≤ T , be the parallel transport in E
along s 7→ (t− s,Xs(a, u)), where
Xs(a, u) = expXs(a√
s//su),
and J−10,s ∇0J0,s = J−1
0,s (0, ·)∇a |a=0 J0,s(a, ·) ∈ T ∗X0M ⊗ End(EX0).
Theorem 7.17 (∇s) is a solution to the parabolic Yang–Mills equation on[0, T )×M if and only if the processes
J−10,s ∇0J0,s, s ∈ [0, t], t < T,
are martingales. The quadratic variation S of J−10,s ∇0J0,s is
Ss =∫ s
0
r| R∇t−r |2(Xr)dr.
Theorem 7.18 (Stochastic characterization of singularities; Sem. Prob. 2003;J. Math. Pures Appl. 2002) Let
Φ(t,x)(s) := E∫ s
s/2
r| R∇t−r |2(Xr(x))dr.
The following conditions are equivalent:
(i) ∇ develops a singularity at (t, x);
(ii) lims→0 Φ(t,x)(s) > 0;
(iii) lims→0 Φ(t,x)(s) ≥ ε0;
(iv) lims→01
log t−log s ‖ J−1s,t ∇0Js,t ‖22> 0
(v) lims→01
log t−log s ‖ J−1s,t ∇0Js,t ‖22≥ (2 log 2)ε0
where ε0 = ε0(E) is a universal constant.
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