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}


\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}


\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


About metric2011

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