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
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Let
be a domain (an open and connected set) in
. Let
be the Laplace operator, let
be a bounded function on the boundary
, 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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/582720b240865c82bfd19df6a1b85acc2e40e5fa)
It can be shown that if a solution
exists, then
is the expected value of
at the (random) first exit point from
for a canonical Brownian motion starting at
. See theorem 3 in Kakutani 1944, p. 710.
The Dirichlet–Poisson problem
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Let
be a domain in
and let
be a semi-elliptic differential operator on
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}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/872c7d20e68b046efc76d56c2c1bc241045ad98f)
where the coefficients
and
are continuous functions and all the eigenvalues of the matrix
are non-negative. Let
and
. 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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29ae4ba5f44d8a82575253027e34641e75aabff1)
The idea of the stochastic method for solving this problem is as follows. First, one finds an Itō diffusion
whose infinitesimal generator
coincides with
on compactly-supported
functions
. For example,
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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f04d55e48fbbdc906d9b63d91e76cbf83dd364)
where
is n-dimensional Brownian motion,
has components
as above, and the matrix field
is chosen so that:
![{\displaystyle {\frac {1}{2}}\sigma (x)\sigma (x)^{\top }=a(x),\quad \forall x\in \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e27be5db6686d9d6c27085db2dc5811707d5ad)
For a point
, let
denote the law of
given initial datum
, and let
denote expectation with respect to
. Let
denote the first exit time of
from
.
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b5dae17bf95e890f8ddb0d01c1504a0639a84b87)
provided that
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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/60f1e4ffcaaca55986d7376b9b3c6ce9dc310355)
It turns out that one further condition is required:
![{\displaystyle \mathbb {P} ^{x}{\big (}\tau _{D}<\infty {\big )}=1,\quad \forall x\in D}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f7d3f55f4ade2d689485f75cc2604aca375568e)
For all
, the process
starting at
almost surely leaves
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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6342ebfa7899b427eebb0290569947b9b7318f5e)
and solves (P1) in the sense that if
denotes the characteristic operator for
(which agrees with
on
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)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d87333399c102bf81990de8df1afb037d745d85)
Moreover, if
satisfies (P2) and there exists a constant
such that, for all
:
![{\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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/158947683823f257050dd86af1c80218de75ce08)
then
.