### 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 **R**^{n}. Let Δ be the Laplace operator, let *g* be a bounded function on the boundary ∂*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 **R**^{n} and let *L* be a semi-elliptic differential operator on *C*^{2}(**R**^{n}; **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 *a*(*x*) = (*a*_{ij}(*x*)) are non-negative. Let *f* ∈ *C*(*D*; **R**) and *g* ∈ *C*(∂*D*; **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* : **R**^{n} → **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 *σ* is chosen so that

- \frac1{2} \sigma (x) \sigma(x)^{\top} = a(x) \mbox{ for all } x \in \mathbf{R}^{n}.

For a point *x* ∈ **R**^{n}, let **P**^{x} denote the law of *X* given initial datum *X*_{0} = *x*, and let **E**^{x} denote expectation with respect to **P**^{x}. Let *τ*_{D} denote the first exit time of *X* from *D*.

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

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

provided that *g* is a bounded function and that

- \mathbf{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:

- \mathbf{P}^{x} \big[ \tau_{D} < + \infty \big] = 1 \mbox{ for all } x \in D,

i.e., 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) = \mathbf{E}^{x} \left[ g \big( X_{\tau_{D}} \big) \right] + \mathbf{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), & \mathbf{P}^{x} \mbox{-a.s., for all } x \in D. \end{cases} \quad \mbox{(P2)}

Moreover, if *v* ∈ *C*^{2}(*D*; **R**) satisfies (P2) and there exists a constant *C* such that, for all *x* ∈ *D*,

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

then *v* = *u*.

## References

- (See Section 9)