Stochastic analysis on manifolds

For the reader's convenience and if not stated otherwise, let (\Omega,\mathcal{A},(\mathcal{F}_t)_{t\geq 0},\mathbb P) be a filtered probability space and M be a smooth manifold. The filtration satisfies the usual conditions, i.e. it is right-continuous and complete. We use the Stratonovich integral which obeys the classical chain rule (compared to Itô calculus). The main advantage for us lies in the fact that stochastic differential equations are then stable under diffeomorphisms f:M\to N between manifolds, i.e. if X is a solution, then also f(X) is a solution under transformations of the stochastic differential equation. Notation: • TM is. the tangent bundle of M. • T^*M is the cotangent bundle of M. • \Gamma(TM) is the C^{\infty}(M)-module of vector fields on M. • X \circ dZ is the Stratonovich integral. • C^{\infty}_c(M) is the space of test functions on M, i.e. f\in C^{\infty}_c(M) is smooth and has compact support. • \widehat{M}:=M\cup \{\infty\} is the one-point compactification (or Alexandroff compactification). == Flow processes ==

Flow processes (also called L-diffusions) are the probabilistic counterpart of integral curves (flow lines) of vector fields. In contrast, a flow process is defined with respect to a second-order differential operator, and thus, generalises the notion of deterministic flows being defined with respect to a first-order operator. Partial differential operator in Hörmander form Let A\in \Gamma(TM) be a vector field, understood as a derivation by the C^{\infty}(M)-isomorphism : \Gamma(TM)\to \operatorname{Der}_{\mathbb{R}} C^{\infty}(M),\quad A\mapsto (f\mapsto Af) for some f\in C^{\infty}(M). The map Af:M\to \mathbb{R} is defined by Af(x):=A_x(f). For the composition, we set A^2:=A(A(f)) for some f\in C^{\infty}(M). A partial differential operator (PDO) L:C^{\infty}(M)\to C^{\infty}(M) is given in Hörmander form if and only there are vector fields A_0,A_1,\dots,A_r\in \Gamma(TM) and L can be written in the form : L=A_0+\sum\limits_{i=1}^r A_i^2. Flow process Let L be a PDO in Hörmander form on M and x\in M a starting point. An adapted and continuous M-valued process X with X_0=x is called a flow process to L starting in x, if for every test function f\in C^{\infty}_c(M) and t\in\mathbb{R}_+ the process : N(f)_t:=f(X_t)-f(X_0)-\int_0^t Lf(X_r)\mathrm{d}r is a martingale, i.e. : \mathbb{E}\left(N(f)_t\mid\mathcal{F}_s\right)=N(f)_s,\quad \forall s\leq t. Remark For a test function f\in C^{\infty}_c(M), a PDO L in Hörmander form and a flow process X_t^x (starting in x) also holds the flow equation, but in comparison to the deterministic case only in mean : \frac{\mathrm{d}}{\mathrm{d}t}\mathbb{E} f(X_t^x) = \mathbb{E}\left[Lf(X_t^x)\right]. and we can recover the PDO by taking the time derivative at time 0, i.e. : \left.\frac{\mathrm{d}}{\mathrm{d}t}\right|_{t=0}\mathbb{E}f(X_t^x)=Lf(x). Lifetime and explosion time Let \empty \neq U\subset \mathbb{R}^n be open und \xi>0 a predictable stopping time. We call \xi the lifetime of a continuous semimartingale X=(X_t)_{0\leq t on U if • there is a sequence of stopping times (\xi_n) with \xi_n\nearrow\xi, such that \xi_n \mathbb P-almost surely on \{0. • the stopped process (X_{t\wedge \xi_n}) is a semimartingale. Moreover, if X_{\xi_n(\omega)}\to \partial U for almost all \omega\in\{\xi, we call \xi explosion time. A flow process X can have a finite lifetime \xi. By this we mean that X=(X)_{t is defined such that if t\to \xi, then \mathbb P-almost surely on \{\xi we have X_t\to \infty in the one-point compactification \widehat{M}:=M\cup \{\infty\}. In that case we extend the process path-wise by X_t:=\infty for t\geq \xi. Semimartingales on a manifold A process X is a semimartingale on M, if for every f\in C^2(M) the random variable f(X) is an \R-semimartingale, i.e. the composition of any smooth function f with the process X is a real-valued semimartingale. It can be shown that any M-semimartingale is a solution of a stochastic differential equation on M. If the semimartingale is only defined up to a finite lifetime \xi, we can always construct a semimartingale with infinite lifetime by a transformation of time. A semimartingale has a quadratic variation with respect to a section in the bundle of bilinear forms on TM. Introducing the Stratonovich Integral of a differential form \alpha along the semimartingale X we can study the so called winding behaviour of X, i.e. a generalisation of the winding number. Stratonovich integral of a 1-form Let X be an M-valued semimartingale and \alpha\in\Gamma(T^*M) be a 1-form. We call the integral \int_X\alpha:=\int \alpha (\circ dX) the Stratonovich integral of \alpha along X. For f\in C^{\infty}(M) we define f(X)\circ \alpha(\circ dX):=f(X)\circ d(\int_X\alpha). == SDEs on a manifold ==

A stochastic differential equation on a manifold M, denoted SDE on M, is defined by the pair (A,Z) including a bundle homomorphism (i.e. a homomorphism of vector bundles) or the (r+1)-tuple (A_1,\dots,A_r,Z) with vector fields A_1,\dots,A_r given. Using the Whitney embedding, we can show that there is a unique maximal solution to every SDE on M with initial condition X_0=x. If we have identified the maximal solution, we recover directly a flow process X^x to the operator L. Definition An SDE on M is a pair (A,Z), where • Z=(Z_t)_{t\in\mathbb{R}_+} is a continuous semimartingale on a finite-dimensional \R-vector space E; and • A:M\times E\to TM is a (smooth) homomorphism of vector bundles over M : A:(x,e)\mapsto A(x)e whereA(x):E\to TM is a linear map. The stochastic differential equation (A,Z) is denoted by : dX=A(X)\circ dZ or : dX=\sum\limits_{i=1}^r A_i(X)\circ dZ^i. The latter follows from setting A_i:=A(\cdot)e_i with respect to a basis (e_i)_{i=1,\dots,r} and \R-valued semimartingales (Z^{i})_{i=1,\dots,r} with Z=\sum\limits_{i=1}^rZ^{i}e_i. As for given vector fields A_1,\dots,A_r\in \Gamma(TM) there is exactly one bundle homomorphism A such that A_i:=A(\cdot)e_i, our definition of an SDE on M as (A_1,\dots,A_r,Z) is plausible. If Z has only finite life time, we can transform the time horizon into the infinite case. Solutions Let (A,Z) be an SDE on M and x_0:\Omega\to M an \mathcal{F}_0-measurable random variable. Let (X_t)_{t be a continuous adapted M-valued process with life time \zeta on the same probability space such as Z. Then (X_t)_{t is called a solution to the SDE : dX=A(X)\circ dZ with initial condition X_0=x_0 up to the life time \zeta, if for every test function f \in C^{\infty}_c(M) the process f(X) is an \R-valued semimartingale and for every stopping time \tau with 0\leq \tau , it holds \mathbb P-almost surely : f(X_\tau) = f(x_0) + \int_0^\tau (df)_{X_s} A(X_s)\circ \mathrm{d}Z_s , where (df)_X:T_xM\to T_{f(x)}M is the push-forward (or differential) at the point X. Following the idea from above, by definition f(X) is a semimartingale for every test function f\in C_c^{\infty}(M), so that X is a semimartingale on M. If the lifetime is maximal, i.e. : \{\zeta \mathbb P-almost surely, we call this solution the maximal solution. The lifetime of a maximal solution X can be extended to all of \R_+ , and after extending f to the whole of \widehat{M}, the equation : f(X_{t})=f(X_0)+\int_0^t (df)_X A(X)\circ dZ, \quad t\geq 0, holdsup to indistinguishability. Remark Let Z=(t,B) with a d-dimensional Brownian motion B=(B_1,\dots,B_d), then we can show that every maximal solution starting in x_0 is a flow process to the operator : L=A_0+\frac{1}{2}\sum\limits_{i=1}^d A_i^2. == Martingales and Brownian motion ==

Brownian motion on manifolds are stochastic flow processes to the Laplace-Beltrami operator. It is possible to construct Brownian motion on Riemannian manifolds (M,g). However, to follow a canonical ansatz, we need some additional structure. Let \mathcal{O}(d) be the orthogonal group; we consider the canonical SDE on the orthonormal frame bundle O(M) over M, whose solution is Brownian motion. The orthonormal frame bundle is the collection of all sets O_x(M) of orthonormal frames of the tangent space T_xM : O(M):=\bigcup\limits_{x\in M}O_x(M) or in other words, the \mathcal{O}(d)-principal bundle associated to TM. Let W be an \R^d-valued semimartingale. The solution U of the SDE : dU_t = \sum\limits_{i=1}^d A_i(U_t)\circ dW_t^i,\quad U_0=u_0, defined by the projection \pi:O(M)\to M of a Brownian motion X on the Riemannian manifold, is the stochastic development from W on M. Conversely we call W the anti-development of U or, respectively, \pi(U)=X. In short, we get the following relations: W\leftrightarrow U \leftrightarrow X, where • U is an O(M)-valued semimartingale; and • X is an M-valued semimartingale. For a Riemannian manifold we always use the Levi-Civita connection and the corresponding Laplace-Beltrami operator \Delta_M. The key observation is that there exists a lifted version of the Laplace-Beltrami operator on the orthonormal frame bundle. The fundamental relation reads, for f\in C^{\infty}(M), : \Delta_M f(x)=\Delta_{O(M)}(f\circ \pi)(u) for all u\in O(M) with \pi u=x, and the operator \Delta_{O(M)} on O(M) is well-defined for so-called horizontal vector fields. The operator \Delta_{O(M)} is called ''Bochner's horizontal Laplace operator''. Martingales with linear connection To define martingales, we need a linear connection \nabla. Using the connection, we can characterise \nabla-martingales, if their anti-development is a local martingale. It is also possible to define \nabla-martingales without using the anti-development. We write \stackrel{m}{=} to indicate that equality holds modulo differentials of local martingales. Let X be an M-valued semimartingale. Then X is a martingale or \nabla-martingale, if and only if for every f\in C^{\infty}(M), it holds that : d(f\circ X)\,\,\stackrel{m}{=}\,\,\tfrac{1}{2}(\nabla df)(dX,dX). Brownian motion on a Riemannian manifold Let (M,g) be a Riemannian manifold with Laplace-Beltrami operator \Delta_{M}. An adapted M-valued process X with maximal lifetime \xi is called a Brownian motion(M,g), if for every f\in C^{\infty}(M) : f(X)-\frac{1}{2}\int\Delta_{M} f(X)\mathrm{d}t is a local \R-martingale with life time \xi. Hence, Brownian motion Bewegung is the diffusion process to \tfrac{1}{2}\Delta_{M}. Note that this characterisation does not provide a canonical procedure to define Brownian motion. == References and notes ==