Brownian Motion on Manifold

Transcription

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

2 Overview 1 Extrinsic construction of Brownian motion Diffusion Process Brownian motion by embedding 2 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Examples of Brownian motion on Riemannian Manifold QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

3 Extrinsic construction of Brownian motion Diffusion Process SDE on manifolds Definition For a differentiable manifold M. An M-valued path x with explosion time e = e(x) > 0 is a continuous map x : [0, ) ˆM such that x t M, for 0 t e and x t = M for all t e if e <. The space of M-valued paths with explosion time is called the path space of M and is denoted by W(M). (W (M), B ) is a filtered measurable space and B t = B t (W (M)) is the σ field algebra. Remark ˆM is the one point compactification of M. semimartingale X t = M t + A t M t : local martingale, A t cadlag adapted process with B.V. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

4 Extrinsic construction of Brownian motion Diffusion Process SDE on manifolds Definition (semimartingale on manifolds ) Let M be a differentiable manifold and (Ω, F, P) a filtered probability space. Let τ be an F -stopping time. A continuous, M-valued process X defined on[0, τ) is called an M-valued semimartingale if f (X ) is a real-valued semimartingale on [0, τ) for all f C (M). QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

5 Extrinsic construction of Brownian motion Diffusion Process SDE on manifolds Definition (semimartingale on manifolds ) Let M be a differentiable manifold and (Ω, F, P) a filtered probability space. Let τ be an F -stopping time. A continuous, M-valued process X defined on[0, τ) is called an M-valued semimartingale if f (X ) is a real-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 a solution of SDE(V 1,..., V l, Z, X 0 ) up to τ if for all f C (M), t f (X t ) = f (X 0 ) + V α f (X s ) dzs α 0 SDE(V 1,..., V l, Z, X 0 ):dx t = V α f (X s ) dz α s QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

6 Extrinsic construction of Brownian motion Diffusion Process Diffusion process Definition (diffusion process) (i)an F -adapted stochastic process X : Ω W (M) defined on a filtered probability space (Ω, F, P) is called a diffusion process generated by L if X is M-valued F semimartingale up to e(x) and t M f (X t ) = f (X t ) f (X 0 ) Lf (X s )ds, 0 t e(x ) 0 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 t M f (ω) t = f (ω t ) f (ω 0 ) Lf (ω s )ds, 0 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, / 26

7 Extrinsic construction of Brownian motion Diffusion Process Diffusion process relation between L-diffusion measure and L-diffusion process X is an L-diffusion υ X = P X 1 is an L diffusion meassure. υ is an L-diffusion measure on W (M) X (ω) t = ω t is an L-diffusion process. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

8 Extrinsic construction of Brownian motion Diffusion Process Diffusion process relation between L-diffusion measure and L-diffusion process X is an L-diffusion υ X = P X 1 is an L diffusion meassure. υ is an L-diffusion measure on W (M) X (ω) t = ω t is an L-diffusion process. local coordinate representation of L L = 1 2 d i,j=1 a i,j (x) x i x j + d i=1 b i (x) x i, where a = a ij ; U S + (d), b = b i : U R d are smooth functions. U is a neighborhood on M covered by the local coordinate system. dx t = b(x t )dt + σ(x t )db t L = b i x i + i,j [σσt ] i,j 2 x i x j QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

9 Extrinsic construction of Brownian motion Diffusion Process Properties about diffusion process Theorem (1.3.4 by Elton P. Hsu) Let L be a smooth second order elliptic operator on a differentiable manifold M and µ 0 a probability measure on M. Then there exists an L-diffusion measure with initial distribution µ 0. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

10 Extrinsic construction of Brownian motion Diffusion Process Properties about diffusion process Theorem (1.3.4 by Elton P. Hsu) Let L be a smooth second order elliptic operator on a differentiable manifold M and µ 0 a probability measure on M. Then there exists an L-diffusion measure with initial distribution µ 0. Theorem (1.3.6 by Elton P. Hsu) An L-diffusion measure with a given initial distribution is unique. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

11 Extrinsic construction of Brownian motion Brownian motion by embedding Brownian motion on euclidean space Brownian motion on R n, transition denssity function: infinitesimal generator: p(t, x, y) = ( 1 2πt )n/2 e x y 2 /2t 1 2 = 1 2 n 2 x 2 i=1 i Note:p(t, x, y) is the smallest positive solution of : p t = 1 2 u, lim p(t, x, y) = δ x(y) t 0 QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

12 Extrinsic construction of Brownian motion Brownian motion by embedding Brownian motion on a Riemannian manifold Definition (Laplace-Beltrami operator) The Laplace-Beltrami operator M on a Riemannian manifold is: M f = div(grad f ) QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

13 Extrinsic construction of Brownian motion Brownian motion by embedding Brownian motion on a Riemannian manifold Definition (Laplace-Beltrami operator) The Laplace-Beltrami operator M on a Riemannian manifold is: M f = div(grad f ) Remark:, x = dsx 2 = g ij dx i dx j : Riemannian metric on M. gradf, X = X (f ), X Γ(M) M X (f )dυ = M fdivxdυ. υ : Riemannian volume measure. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

14 Extrinsic construction of Brownian motion Brownian motion by embedding Brownian motion on a Riemannian manifold Definition (Laplace-Beltrami operator) The Laplace-Beltrami operator M on a Riemannian manifold is: M f = div(grad f ) Remark:, x = dsx 2 = g ij dx i dx 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 M f = 1 G ( Gg ij f ) = g ij x j x i 2 f x i x j + b i f, b i = 1 ( Gg x i ij ) G x j QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

15 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion Stratonovich integral A general SDE in Stratonovich form: dx t = V α (X ) t dwt α + V 0 (X t )dt It generates a diffusion process with infinitesimal generator : L = 1 l 2 i=1 V α 2 + V 0 QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

16 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion Stratonovich integral A general SDE in Stratonovich form: dx t = V α (X ) t dwt α + V 0 (X t )dt It generates a diffusion process with infinitesimal generator : L = 1 l 2 i=1 V α 2 + V 0 Theorem (Whitney s embedding theorem) Suppose that M is a differentiable manifold. Then there exists an embedding i : M R N for some N such that the image i(m) is a closed subset of R N.(N = 2 dim M + 1 will do) QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

17 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion Stratonovich integral A general SDE in Stratonovich form: dx t = V α (X ) t dwt α + V 0 (X t )dt It generates a diffusion process with infinitesimal generator : L = 1 l 2 i=1 V α 2 + V 0 Theorem (Whitney s embedding theorem) Suppose that M is a differentiable manifold. Then there exists an embedding i : M R N for some N such that the image i(m) is a closed subset of R N.(N = 2 dim M + 1 will do) Theorem (Nash s embedding theorem) Every Riemannian manifold can be isometrically embedded in some euclidean space with the standard metric. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

18 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion square representation of M M = lα P2 α QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

19 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion square representation of M M = lα P2 α P α :orthogonal projection : ξ α T x M. vector field on M. {ξ α } : standard orthonormal basis on R l, x M, QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

20 Extrinsic construction of Brownian motion Brownian motion by embedding extrinsic construction of Brownian motion square representation of M M = lα P2 α P α :orthogonal projection : ξ α T x M. vector field on M. {ξ α } : standard orthonormal basis on R l, x M, proof file up later Brownian motion by embedding Consider Stratonovich SDE on M driven by a l-dim euclidean Brownian motion W: dx t = P α (X t ) dw α t, X 0 M. The solution is a diffusion process generated by 1 2 l α=1 P2 α = 1 2 M. Any M-valued diffusion process generated by M /2 is called a Brownian motion on M. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

21 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. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

22 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 and Laplave-Beltrami operator M. Given a probability measure µ,! M /2-diffusion measure P µ on (W (M), B ). Any M /2-diffusion measure on W(M) is called a Wiener measure on W(M). Roughly speaking: Brownian motion on M is any M-valued stochastic process X whose law is a Wiener measure on the path space W(M). QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

23 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion differential geometry concepts connection x T x M tangent space TM tangent bundle Γ(TM) A connection : Γ(TM) Γ(TM) Γ(TM) satisfies: 1) fx +gy Z = f X Z + g Y Z,, 2) X (Y + Z) = X Y + X Z 3) X (fy ) = f X Y + X (f )Y X Y : covariant differentiation to Y along X. A vector field V along a curve {x t } on M is parallel along the curve if ẋv = 0 at each point of the curve. frame bundle A frame at x is R linear isomorphism u : R d T x M, e i ue i The frame bundle :F (M) = x M F (M) x F (M) x : space of all frames at x, QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

24 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Horizontal lift and stochastic development cannonical projection π : F (M) M A tangent vector X T u F (M) is called vertical if it is tangent to the fiber F (M) πu. V u F (M) : space of vertical vectors. The curve {u t } is horizontal if each e R d, the the vector field {u t e} is parallel along {πu t }. A tangent vector X T u F (M) is horizontal if it is the tangent vector of a horizontal curve form u. We have the relation: T u F (M) = V u F (M) H u F (M) If we consider the orthonormal frames of the tangent space : T u O(M) = H u O(M) V u O(M) QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

25 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Horizontal lift and stochastic development The projection π : O(M) M induces an isomorphism π : H u O(M) T x M. At each u O(M) H i is the unique horizontal vector in H u O(M) whose projection is the ith unit vector ue i of the orthonormal frame: π H i (u) = ue i, H i (u) H u O(M) H i (u) is the horizontal lift of ue i.(in general, e R d ) Bochner s horizontal Laplacian O(M) = n i=1 H2 i is called the Bochner s horizontal Laplacian on O(M) Important relation M f (X ) = O(M) (f π)(u), u O(M) with πu = x. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

26 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion horizontal lift and stochastic development Let {u t } be a horizontal lift of differentiable curve {x t } on M. since x t T xt M, so ut 1 x t R d. The anti-development of the curve {x t } is a curve {ω t } in R d defined by: ω t = t us 1 0 x s ds The anti-development {ω t } and the horizontal lift {u t } of a curve {x t } are connected by an ODE: u t = H i (u t ) ω t i QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

27 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Horizontal lift and stochastic development u t = H i (u t ) ω t i du t = d i=1 H i(u t ) dwt i. Definition An F(M)-valued semimartingale U is said to be horizontal if there exists an R d valued semimartingale W such that the above ODE holds. The unique W is called the anti-development of U(or of its projectionx = πu) Ler W be an R d valued semimartingale and U 0 an F(M)-valued, F 0 measurable random variable. The solution U of the above SDE is called a stochastic development of W in F(M). Its projection X = πu is called a stochastic development of W in M. Let X be an M-valued semimartingale. An F(M)-valued horizontal semimartingale such that its projection πu = X is called a horizontal lift of X. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

28 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Horizontal lift and stochastic development W U X Lemma Suppose that M is a closed submanifold of R N. For each x M, let P(x) : R N T x M be the orthogonal projection form R N to the tangent space T x M. If X is an M-valued semimartingale, then X t = X 0 + t 0 P(X s) dx s. Theorem A horizontal semimartingale U on the frame bundle F(M) has a unique anti-development W. In fact, W t = t 0 U 1 s P α (X s ) dxs α, where X t = πu t. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

29 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion horizontal lift and stochastic lift Theorem Suppose that X = {X t, 0 t τ} is a semimartingale on M up to stopping tie τ, and U 0 an F(M)-valued F 0 -random variable such that πu 0 = X 0. Then there is a unique horizontal lift {U t, 0 t τ} of X starting from U 0. Proposition Let a semimartingale X on a manifold M be the solution of SDE(V 1,, V N ; Z, X 0 ) and let Vα be the horizontal lift of V α to the frame bundle F(M). Then the horiaontal lift U of X is the solution of SDE(V!,, V N ; Z, U 0), and the anti-development of X is given by W t = t 0 U 1 s V α (X s ) dzs α. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

30 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion 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 X0 1 be its initial distribution. Then the following statements are equivalent. X is a M /2-diffusion process(a solution to the martingale problem for M /2 with respect to its own filtration F X ), i.e., M f (X ) t := f (X t ) f (X 0 ) Mf (X s )ds, 0 t e(x ), is an F X -local martingale for all f C (M). The law P X = P X 1 is a Wiener measure on (W (M), B(W (M))), i.e., P X = P µ. X is a F X -semimartingale on M whose anti-development is a standard euclidean Brownian motion. An M-valued process X satisfying any of the above conditions is called a (Riemannian) Brownian motion on M. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

31 Intrinsic construction of Brownian motion Ells-Elworthy-Malliavin construction of Brownian motion Horizontal Brownian motion Proposition Let U : Ω W (O(M)) be a measurable map defined on a probability space (Ω, F, P). Then the following statements are equivalent. U is a O(M) /2 diffusion with respect to its own filtration F U. U is a horizontal F U semimartingale on M whose projection X = πu is a Brownian motion on M. U is a horizontal F U -semimartingale on M whose anti-development is a standard euclidean Brownian motion. An O(M)-valued process U satisfying any fo the above condition is called a horizontal Brownian motion on O(M). QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

32 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold Brownian motion on a circle Consider the compact manifold: S 1 = {e iθ : 0 θ 2π} R 2 W: a Brownian motion on R 1. Then The Brownian motion on S 1 is given by X t = e iwt. The anti-development of X is W. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

33 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold Brownian motion on a sphere Consider the d-sphere embedded in R d + 1. S d = {x R d+1 : x 2 = 1} The projection to the tangent sphere at x is given by P(x)ξ = ξ ξ, x x, x S d, ξ R d+1 then the matrix P(x) is P(x) ij = δ ij x i x j. The Brownian motion on S d is the solution of the equation : X i t = X i 0 + t 0 (δ ij X i s X j s ) dw j s, X 0 S d. This is called the stroock s representation of spherical Brownian motion. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

34 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold Brownian motion on a radially symmetric manifold Consider a radially symmetric manifold.in the polar coordinates (r, θ) R + S d+1 determined by the exponential map at its pole, the metric has a special form: ds 2 = dr 2 + G(r) 2 dθ 2, where dθ 2 : Riemannian metric on S d 1, G is a smooth function on an interval [0,D) with G(0) = 0, G (0) = 1. The Laplace-beltrami operator has the form: Where L r is the radial Laplacian M = L r + 1 G(r) 2 S d 1. L r = ( r )2 + (d 1) G (r) G(r) r. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

35 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold Brownian motion on a radially symmetric manifold Let X t = (r t, θ t ) be a Brownian motion on a radially symmetric manifold M in polar coordinates. We can construct a Brownian motion on a radially symmetric manifold as a warped product. {r t }: a diffusion process generated by the radial Laplacian L r {z t }: an independent Brownian motion on d 1. By some rescaling, t X t = (r t, z lt ) is a Brownian motion on the radially symmetric manifold. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

36 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold Brownian motion on a radially symmetric manifold Let X t = (r t, θ t ) be a Brownian motion on a radially symmetric manifold M in polar coordinates. We can construct a Brownian motion on a radially symmetric manifold as a warped product. {r t }: a diffusion process generated by the radial Laplacian L r {z t }: an independent Brownian motion on d 1. By some rescaling, t X t = (r t, z lt ) is a Brownian motion on the radially symmetric manifold. Using martingale property of the Brownian motion X to the distance function f (r, θ) = r we have r t = r 0 + β t + d 1 2 For angular process, consider f C (S d 1 ). Then f (θ t ) = f (θ 0 ) + M f t f (z t ) = f (z 0 ) + M f τ t t 0 t 0 G (r s) G(r s) ds. S d 1 f (θ s) G(r s) ds,scaling by l 2 t = t 0 ds G(r s) 2. We have t 0 S d 1f (z s)ds, where z t = θ τt, τ t inverse of l t. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26

37 Intrinsic construction of Brownian motion Examples of Brownian motion on Riemannian Manifold The End This slides is mainly based on first three chapters of Elton Hsu s book Analysis on Manifolds. QI FENG (Purdue University) Brownian Motion on Manifold August 31, / 26