The geometry of subelliptic diffusions
1. Stochastic flows
Let be a vectorfield with flow . For compactly supported functions ,
Can one attach a flow to a second order operator ? E.g. to
Basic example is the Euclidean Laplacian. Answer is yes, but flow lines now depend on a random parameter , . Also, they are no more differentiable as function of . In other words, the flow becomes a stochastic process .
1.1. Formal definition
The data are a filtered probability space, i.e. a probability space equipped with an increasing -algebra . Think of as representing the events having occurred up to time .
An adapted continuous process is a family of random variables , -measurable, with a.e. continuous trajectories. It is a flow process of , (or an -diffusion) with starting point at if and, for compactly supported functions ,
is a martingale, i.e. for every , the conditional expectation
In other words, has no specific trend, it is only fluctuations. One can also say that is the best possible prediction of one can do with the knowledge one has at time .
The operator is a semi-group. Since , the flow law implies that
In particular, the governing differential operator is recovered by
Remark. The life time of the process may be finite. It is a stopping time . We assume that implies that tends to infinity. Then, for functions which need not be compactly supported, is a local martingale, i.e. it becomes a martingale when stopped at .
1.2. Basic example
Let be half the Laplacian. The corresponding flow is the Brwnian motion . It\^o’s formula states that
The first term is a stochastic integral. It is a martingale.
What is this good for ? Here are a few applications.
1.3. The Dirichlet problem
The problem: let be an open set, a continuous function on the boundary . Find a continuous extension to which is -harmonic, i.e. .
Assume that there exists an -diffusion with a.s. finite life time. Assume is a solution to the Dirichlet problem. Exhaust with compact sets . Let be a compactly supported function that coincides with on . Let denote the exit time from . Then, for ,
and taking a limits, first to infinity, then to infinity,
since, by assumption, a.s., i.e. a.s.
In other words, we get uniqueness of the classical solution of Dirichlet’s problem under the single condition a.s. It also leads to an efficient numerical method for computing the solution, the Monte-Carlo method.
Conversely, define a function by . It is true that is and . In order to prove that extends continuously to , one needs that tends to 0 in probability as tends to a point of .
Example 1 on annulus has no uniqueness.
Indeed, any radial function vanishing on the boundary is -harmonic. In fact, is 1-dimensional Brownian motion on each circle. It never exists, so .
Example 2 on a symmetric bean shaped planar domain has no existence.
With a boundary data symmetric on the convex part, any solution takes values determined by the convex boundary. Indeed, does not tend to 0 as tends to the point where the concave boundary touches the axis.
1.4. Heat equation