In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, some
boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to t ...
s 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
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
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
be a domain (an
open and
connected set
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that ...
) in
. Let
be the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
, let
be a
bounded function
In mathematics, a function ''f'' defined on some set ''X'' with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number ''M'' such that
:, f(x), \le M
for all ''x'' in ''X''. A fun ...
on the
boundary , and consider the problem:
:
It can be shown that if a solution
exists, then
is the
expected value
In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
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
Let
be a domain in
and let
be a semi-elliptic differential operator on
of the form:
:
where the coefficients ''
'' and ''
'' are
continuous function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s and all the
eigenvalues of the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
''
'' are non-negative. Let ''
'' and ''
''. Consider the
Poisson problem
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with th ...
:
:
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:
:
where
is ''n''-dimensional Brownian motion, ''
'' has components ''
'' as above, and the
matrix field
In abstract algebra, a matrix field is a field with matrices as elements. In field theory there are two types of fields: finite fields and infinite fields. There are several examples of matrix fields of different characteristic and cardinality.
...
''
'' is chosen so that:
:
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:
: