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 ${\displaystyle D}$ be a domain (an open and connected set) in ${\textstyle \mathbb {R} ^{n}}$. Let ${\displaystyle \Delta }$ be the Laplace operator, let ${\displaystyle g}$ be a bounded function on the boundary ${\displaystyle \partial D}$, and consider the problem:

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

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

The Dirichlet–Poisson problem

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

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

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

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

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

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

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

${\displaystyle {\frac {1}{2}}\sigma (x)\sigma (x)^{\top }=a(x),\quad \forall x\in \mathbb {R} ^{n}}$

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

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

${\displaystyle u(x)=\mathbb {E} ^{x}\left[g{\big (}X_{\tau _{D}}{\big )}\cdot \chi _{\{\tau _{D}<+\infty \}}\right]+\mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}f(X_{t})\,\mathrm {d} t\right]}$

provided that ${\displaystyle g}$ is a bounded function and that:

${\displaystyle \mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}{\big |}f(X_{t}){\big |}\,\mathrm {d} t\right]<+\infty }$

It turns out that one further condition is required:

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

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

${\displaystyle u(x)=\mathbb {E} ^{x}\left[g{\big (}X_{\tau _{D}}{\big )}\right]+\mathbb {E} ^{x}\left[\int _{0}^{\tau _{D}}f(X_{t})\,\mathrm {d} t\right]}$

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

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

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

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

then ${\displaystyle v=u}$.

References

• Kakutani, Shizuo (1944). "Two-dimensional Brownian motion and harmonic functions". Proc. Imp. Acad. Tokyo. 20 (10): 706–714. doi: 10.3792/pia/1195572706 .
• Kakutani, Shizuo (1944). "On Brownian motions in n-space". Proc. Imp. Acad. Tokyo. 20 (9): 648–652. doi: 10.3792/pia/1195572742 .
• Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN   3-540-04758-1. (See Section 9)