Stochastic processes and boundary value problems

In mathematics, some boundary value problems can be solved using the methods of stochastic analysis. Perhaps the most celebrated example is Shizuo Kakutani's 1944 solution of the Dirichlet problem for the Laplace operator using Brownian motion. However, it turns out that for a large class of semi-elliptic second-order partial differential equations the associated Dirichlet boundary value problem can be solved using an Itō process that solves an associated stochastic differential equation.

Introduction: Kakutani's solution to the classical Dirichlet problem
Let $$D$$ be a domain (an open and connected set) in $\mathbb{R}^{n}$. Let $$\Delta$$ be the Laplace operator, let $$g$$ be a bounded function on the boundary $$\partial D$$, and consider the problem:


 * $$\begin{cases} - \Delta u(x) = 0, & x \in D \\ \displaystyle{\lim_{y \to x} u(y)} = g(x), & x \in \partial D \end{cases}$$

It can be shown that if a solution $$u$$ exists, then $$u(x)$$ is the expected value of $$g(x)$$ at the (random) first exit point from $$D$$ for a canonical Brownian motion starting at $$x$$. See theorem 3 in Kakutani 1944, p. 710.

The Dirichlet–Poisson problem
Let $$D$$ be a domain in $\mathbb{R}^{n}$ and let $$L$$ be a semi-elliptic differential operator on $C^{2}(\mathbb{R}^{n};\mathbb{R})$  of the form:


 * $$L = \sum_{i = 1}^{n} b_{i} (x) \frac{\partial}{\partial x_{i}} + \sum_{i, j = 1}^{n} a_{ij} (x) \frac{\partial^{2}}{\partial x_{i} \, \partial x_{j}}$$

where the coefficients $$b_{i}$$ and $$a_{ij}$$ are continuous functions and all the eigenvalues of the matrix $$\alpha(x) = a_{ij}(x)$$ are non-negative. Let $f\in C(D;\mathbb{R})$  and $g\in C(\partial D;\mathbb{R})$ . Consider the Poisson problem:


 * $$\begin{cases} - L u(x) = f(x), & x \in D \\ \displaystyle{\lim_{y \to x} u(y)} = g(x), & x \in \partial D \end{cases} \quad \mbox{(P1)}$$

The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion $$X$$ whose infinitesimal generator $$A$$ coincides with $$L$$ on compactly-supported $$C^{2}$$ functions $$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$$. For example, $$X$$ can be taken to be the solution to the stochastic differential equation:


 * $$\mathrm{d} X_{t} = b(X_{t}) \, \mathrm{d} t + \sigma (X_{t}) \, \mathrm{d} B_{t}$$

where $$B$$ is n-dimensional Brownian motion, $$b$$ has components $$b_{i}$$ as above, and the matrix field $$\sigma$$ is chosen so that:


 * $$\frac1{2} \sigma (x) \sigma(x)^{\top} = a(x), \quad \forall x \in\mathbb{R}^{n}$$

For a point $$x\in\mathbb{R}^{n}$$, let $$\mathbb{P}^{x}$$ denote the law of $$X$$ given initial datum $$X_{0} = x$$, and let $$\mathbb{E}^{x}$$denote expectation with respect to $$\mathbb{P}^{x}$$. Let $$\tau_{D}$$ denote the first exit time of $$X$$ from $$D$$.

In this notation, the candidate solution for (P1) is:


 * $$u(x) = \mathbb{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \cdot \chi_{\{ \tau_{D} < + \infty \}} \right] + \mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]$$

provided that $$g$$ is a bounded function and that:


 * $$\mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} \big| f(X_{t}) \big| \, \mathrm{d} t \right] < + \infty$$

It turns out that one further condition is required:


 * $$\mathbb{P}^{x} \big( \tau_{D} < \infty \big) = 1, \quad \forall x \in D$$

For all $$x$$, the process $$X$$ starting at $$x$$ almost surely leaves $$D$$ in finite time. Under this assumption, the candidate solution above reduces to:


 * $$u(x) = \mathbb{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \right] + \mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]$$

and solves (P1) in the sense that if $$\mathcal{A}$$ denotes the characteristic operator for $$X$$ (which agrees with $$A$$ on $$C^{2}$$ functions), then:


 * $$\begin{cases} - \mathcal{A} u(x) = f(x), & x \in D \\ \displaystyle{\lim_{t \uparrow \tau_{D}} u(X_{t})} = g \big( X_{\tau_{D}} \big), & \mathbb{P}^{x} \mbox{-a.s.,} \; \forall x \in D \end{cases} \quad \mbox{(P2)}$$

Moreover, if $v \in C^{2}(D;\mathbb{R})$ satisfies (P2) and there exists a constant $$C$$ such that, for all $$x\in D$$:


 * $$| v(x) | \leq C \left( 1 + \mathbb{E}^{x} \left[ \int_{0}^{\tau_{D}} \big| g(X_{s}) \big| \, \mathrm{d} s \right] \right)$$

then $$v=u$$.