In mathematics, stochastic analysis on manifolds or stochastic differential geometry is the study of stochastic analysis over smooth manifolds. It is therefore a synthesis of stochastic analysis (the extension of calculus to stochastic processes) and of differential geometry. The connection between analysis and stochastic processes stems from the fundamental relation that the infinitesimal generator of a continuous strong Markov process is a second-order elliptic operator. The infinitesimal generator of Brownian motion is the Laplace operator and the transition probability density p ( t , x , y ) {\displaystyle p(t,x,y)} of Brownian motion is the minimal heat kernel of the heat equation. Interpreting the paths of Brownian motion as characteristic curves of the operator, Brownian motion can be seen as a stochastic counterpart of a flow to a second-order partial differential operator. Stochastic analysis on manifolds investigates stochastic processes on non-linear state spaces or manifolds. Classical theory can be reformulated in a coordinate-free representation. In that, it is often complicated (or not possible) to formulate objects with coordinates of R d {\displaystyle \mathbb {R} ^{d}} . Thus, we require an additional structure in form of a linear connection or Riemannian metric to define martingales and Brownian motion on manifolds. Therefore, controlled by the Riemannian metric, Brownian motion will be a local object by definition. However, its stochastic behaviour determines global aspects of the topology and geometry of the manifold. Brownian motion is defined to be the diffusion process generated by the Laplace-Beltrami operator 1 2 Δ M {\displaystyle {\tfrac {1}{2}}\Delta _{M}} with respect to a manifold M {\displaystyle M} and can be constructed as the solution to a non-canonical stochastic differential equation on a Riemannian manifold. As there is no Hörmander representation of the operator Δ M {\displaystyle \Delta _{M}} if the manifold is not parallelizable, i.e. if the tangent bundle is not trivial, there is no canonical procedure to construct Brownian motion. However, this obstacle can be overcome if the manifold is equipped with a connection: We can then introduce the stochastic horizontal lift of a semimartingale and the stochastic development by the so-called Eells-Elworthy-Malliavin construction. The latter is a generalisation of a horizontal lift of smooth curves to horizontal curves in the frame bundle, such that the anti-development and the horizontal lift are connected by a stochastic differential equation. Using this, we can consider an SDE on the orthonormal frame bundle of a Riemannian manifold, whose solution is Brownian motion, and projects down to the (base) manifold via stochastic development. A visual representation of this construction corresponds to the construction of a spherical Brownian motion by rolling without slipping the manifold along the paths (or footprints) of Brownian motion left in Euclidean space. Stochastic differential geometry provides insight into classical analytic problems, and offers new approaches to prove results by means of probability. For example, one can apply Brownian motion to the Dirichlet problem at infinity for Cartan-Hadamard manifolds or give a probabilistic proof of the Atiyah-Singer index theorem. Stochastic differential geometry also applies in other areas of mathematics (e.g. mathematical finance). For example, we can convert classical arbitrage theory into differential-geometric language (also called geometric arbitrage theory).
Preface
For the reader's convenience and if not stated otherwise, let ( Ω , A , ( F t ) t ≥ 0 , P ) {\displaystyle (\Omega ,{\mathcal {A}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )} be a filtered probability space and M {\displaystyle 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 → N {\displaystyle f:M\to N} between manifolds, i.e. if X {\displaystyle X} is a solution, then also f ( X ) {\displaystyle f(X)} is a solution under transformations of the stochastic differential equation. Notation:
T M {\displaystyle TM} is. the tangent bundle of M {\displaystyle M} .
T ∗ M {\displaystyle T^{*}M} is the cotangent bundle of M {\displaystyle M} .
Γ ( T M ) {\displaystyle \Gamma (TM)} is the C ∞ ( M ) {\displaystyle C^{\infty }(M)} -module of vector fields on M {\displaystyle M} .
X ∘ d Z {\displaystyle X\circ dZ} is the Stratonovich integral.
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