## Notes of Anton Thalmaier’s lecture nr 1

The geometry of subelliptic diffusions

1. Stochastic flows

Let ${A}$ be a vectorfield with flow ${(\phi_t)}$. For compactly supported functions ${f}$,

$\displaystyle \begin{array}{rcl} f\circ\phi_t(x)-f(x)-\int_{0}^{t}A(f(\phi_s(x)))\,ds=0. \end{array}$

Can one attach a flow to a second order operator ? E.g. to

$\displaystyle \begin{array}{rcl} L=A_0+\sum A_i^2. \end{array}$

Basic example is the Euclidean Laplacian. Answer is yes, but flow lines now depend on a random parameter ${\omega}$, ${\phi_t(x,\omega)}$. Also, they are no more differentiable as function of ${t}$. In other words, the flow becomes a stochastic process ${X_t(x)=\phi_t(x,\omega)}$.

1.1. Formal definition

The data are a filtered probability space, i.e. a probability space ${(\Omega,\mathcal{F},P)}$ equipped with an increasing ${\sigma}$-algebra ${\mathcal{F}_t\subset\mathcal{F}}$. Think of ${\mathcal{F}_t}$ as representing the events having occurred up to time ${t}$.

An adapted continuous process is a family of random variables ${X_t(x)}$, ${\mathcal{F}_t}$-measurable, with a.e. continuous trajectories. It is a flow process of ${L}$, (or an ${L}$-diffusion) with starting point at ${x}$ if ${X_0(x)=x}$ and, for compactly supported functions ${f}$,

$\displaystyle \begin{array}{rcl} N_t^f(x):=f(X_t(x))-f(x)-\int_{0}^{t}(Lf)(X_s(x)))\,ds \end{array}$

is a martingale, i.e. for every ${s\leq t}$, the conditional expectation

$\displaystyle \begin{array}{rcl} \mathop{\mathbb E}(N_t^f(x)-N_s^f(x)|\mathcal{F}_s)=0. \end{array}$

In other words, ${N_t^f}$ has no specific trend, it is only fluctuations. One can also say that ${N_s^f(x)}$ is the best possible prediction of ${N_t^f(x)}$ one can do with the knowledge one has at time ${s}$.

The operator ${f\mapsto P_t(f)=\mathop{\mathbb E}(f(X_t))}$ is a semi-group. Since ${\mathop{\mathbb E}(N_t^f(x))=\mathop{\mathbb E}(N_0^f(x))=0}$, the flow law implies that

$\displaystyle \begin{array}{rcl} \frac{d}{dt}P_t (f)=P_t(Lf). \end{array}$

In particular, the governing differential operator ${L}$ is recovered by

$\displaystyle \begin{array}{rcl} \frac{d}{dt}P_t (f)_{|t=0}=Lf. \end{array}$

Remark. The life time of the process may be finite. It is a stopping time ${\zeta(x)}$. We assume that ${\zeta(x)=\infty}$ implies that ${X_t(x)}$ tends to infinity. Then, for functions ${f}$ which need not be compactly supported, ${N_t^f(x)}$ is a local martingale, i.e. it becomes a martingale when stopped at ${\zeta(x)}$.

1.2. Basic example

Let ${L=\frac{1}{2}\Delta}$ be half the Laplacian. The corresponding flow is the Brwnian motion ${B_t}$. It\^o’s formula states that

$\displaystyle \begin{array}{rcl} f(X_t)-f(X_0)+\int_{0}^{t}\langle\nabla f(X_s),dX_s\rangle +\frac{1}{2}\int_{0}^{t}\Delta f(X_s)\,ds. \end{array}$

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 ${D}$ be an open set, ${\phi}$ a continuous function on the boundary ${\partial D}$. Find a continuous extension ${u}$ to ${\bar{D}}$ which is ${L}$-harmonic, i.e. ${Lu=0}$.

Assume that there exists an ${L}$-diffusion ${X_t}$ with a.s. finite life time. Assume ${u}$ is a solution to the Dirichlet problem. Exhaust ${D}$ with compact sets ${D_n}$. Let ${u_n}$ be a compactly supported function that coincides with ${u}$ on ${D_n}$. Let ${\tau_n(x)}$ denote the exit time from ${D_n}$. Then, for ${x\in D_n}$,

$\displaystyle \begin{array}{rcl} N_t(x):=u_n(X_t(x))-u_n(x)-\int_{0}^{t}(Lu_n)(X_s(x)))\,ds \end{array}$

satisfies

$\displaystyle \begin{array}{rcl} 0=\mathop{\mathbb E}(N_{t\wedge \tau_n(x)}(x))=\mathop{\mathbb E}(u_n(X_{t\wedge \tau_n(x)}(x)))-\mathop{\mathbb E}(u_n(x))=\mathop{\mathbb E}(u(X_{t\wedge \tau_n(x)}(x)))-u(x). \end{array}$

Thus

$\displaystyle \begin{array}{rcl} u(x)=\mathop{\mathbb E}(u(X_{t\wedge\tau_n(x)}(x))). \end{array}$

and taking a limits, first ${n}$ to infinity, then ${t}$ to infinity,

$\displaystyle \begin{array}{rcl} u(x)=\mathop{\mathbb E}(u(X_{\tau(x)}(x)))=\mathop{\mathbb E}(\phi(X_{\tau(x)}(x))), \end{array}$

since, by assumption, ${\tau(x)<\infty}$ a.s., i.e. ${\phi(X_{\tau(x)}(x))\in\partial D}$ a.s.

In other words, we get uniqueness of the classical solution of Dirichlet’s problem under the single condition ${\tau(x)<\infty}$ a.s. It also leads to an efficient numerical method for computing the solution, the Monte-Carlo method.

Conversely, define a function ${u}$ by ${u(x)=\mathop{\mathbb E}(\phi(X_{\tau(x)}(x)))}$. It is true that ${u}$ is ${C^2}$ and ${Lu=0}$. In order to prove that ${u}$ extends continuously to ${\partial D}$, one needs that ${\tau(x)}$ tends to 0 in probability as ${x}$ tends to a point of ${\partial D}$.

Example 1 ${L=\partial_{\theta}^2}$ on annulus has no uniqueness.

Indeed, any radial function vanishing on the boundary is ${L}$-harmonic. In fact, ${X_t}$ is 1-dimensional Brownian motion on each circle. It never exists, so ${\tau\equiv\infty}$.

Example 2 ${L=\partial_{x}^2}$ on a symmetric bean shaped planar domain ${D}$ has no existence.

With a boundary data symmetric on the convex part, any solution takes values determined by the convex boundary. Indeed, ${\tau(x)}$ does not tend to 0 as ${x}$ tends to the point where the concave boundary touches the ${x}$ axis.

1.4. Heat equation