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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 D 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 \mathbb^. Let \Delta 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 g 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 \partial D, and consider the problem: :\begin - \Delta u(x) = 0, & x \in D \\ \displaystyle = g(x), & x \in \partial D \end It can be shown that if a solution u exists, then u(x) 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 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^ and let L be a semi-elliptic differential operator on C^(\mathbb^;\mathbb) of the form: :L = \sum_^ b_ (x) \frac + \sum_^ a_ (x) \frac where the coefficients ''b_'' and ''a_'' 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 ...
''\alpha(x) = a_(x)'' are non-negative. Let ''f\in C(D;\mathbb)'' and ''g\in C(\partial D;\mathbb)''. 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 ...
: :\begin - L u(x) = f(x), & x \in D \\ \displaystyle = g(x), & x \in \partial D \end \quad \mbox 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^ functions f:\mathbb^\rightarrow \mathbb. For example, X can be taken to be the solution to the stochastic differential equation: :\mathrm X_ = b(X_) \, \mathrm t + \sigma (X_) \, \mathrm B_ where B is ''n''-dimensional Brownian motion, ''b'' has components ''b_'' 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. ...
''\sigma'' is chosen so that: :\frac1 \sigma (x) \sigma(x)^ = a(x), \quad \forall x \in\mathbb^ For a point x\in\mathbb^, let \mathbb^ denote the law of X given initial datum X_ = x, and let \mathbb^denote expectation with respect to \mathbb^. Let ''\tau_'' denote the first exit time of X from D. In this notation, the candidate solution for (P1) is: :u(x) = \mathbb^ \left g \big( X_ \big) \cdot \chi_ \right+ \mathbb^ \left \int_^ f(X_) \, \mathrm t \right/math> provided that g is 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 ...
and that: :\mathbb^ \left f(X_) \big, \, \mathrm t \right< + \infty It turns out that one further condition is required: :\mathbb^ \big( \tau_ < \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^ \left g \big( X_ \big) \right+ \mathbb^ \left \int_^ f(X_) \, \mathrm t \right/math> and solves (P1) in the sense that if \mathcal denotes the characteristic operator for X (which agrees with A on C^ functions), then: :\begin - \mathcal u(x) = f(x), & x \in D \\ \displaystyle = g \big( X_ \big), & \mathbb^ \mbox \; \forall x \in D \end \quad \mbox Moreover, if v \in C^(D;\mathbb) satisfies (P2) and there exists a constant C such that, for all x\in D: :, v(x) , \leq C \left( 1 + \mathbb^ \left g(X_) \big, \, \mathrm s \right\right) then v=u.


References

* * * {{cite book , last = Øksendal , first = Bernt K. , authorlink = Bernt Øksendal , title = Stochastic Differential Equations: An Introduction with Applications , edition = Sixth , publisher=Springer , location = Berlin , year = 2003 , isbn = 3-540-04758-1 (See Section 9) Boundary value problems Partial differential equations Stochastic differential equations