Stochastic analysis on manifolds

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

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.

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

and can be constructed as the solution to a non-canonical stochastic differential equation on a Riemannian manifold.

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.

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.

[3] 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[4] or give a probabilistic proof of the Atiyah-Singer index theorem.

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

-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.

) also holds the flow equation, but in comparison to the deterministic case only in mean and we can recover the PDO by taking the time derivative at time 0, i.e. Let

, 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

Using the Whitney embedding, we can show that there is a unique maximal solution to every SDE on

If we have identified the maximal solution, we recover directly a flow process

be the orthogonal group; we consider the canonical SDE on the orthonormal frame bundle

, where For a Riemannian manifold we always use the Levi-Civita connection and the corresponding Laplace-Beltrami operator

The key observation is that there exists a lifted version of the Laplace-Beltrami operator on the orthonormal frame bundle.

is called Bochner's horizontal Laplace operator.

to indicate that equality holds modulo differentials of local martingales.

Note that this characterisation does not provide a canonical procedure to define Brownian motion.