Stochastic processes and boundary value problems explained

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 $\backslash mathbb^$. Let \Delta be the Laplace operator, let g be a bounded function on the boundary \partial D , and consider the problem:\begin{cases} - \Delta u(x) = 0 , & x \in D \ \displaystyle{\lim _{y} 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 $\backslash mathbb^$ and let L be a semi-elliptic differential operator on $C^(\backslash mathbb^;\backslash mathbb)$ of the form:L = \sum ^{n} _{i =1} b _{i} (x) (\partial)/(\partial x _{i}) + \sum ^{n} _{i, j =1} a _{ij} (x) (\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 \alpha(x) = a _{ij} (x) are non-negative. Let $f\backslash in\; C(D;\backslash mathbb)$ and $g\backslash in\; C(\backslash partial\; D;\backslash mathbb)$. Consider the Poisson problem:\begin{cases} - L u(x) = f(x), & x \in D \ \displaystyle{\lim _{y} u(y)} = g(x), & x \in \partial D \end{cases} (P1) 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:d X _{t} = b(X _{t} ) d t + \sigma (X _{t} ) d B _{t}

where

B is n-dimensional Brownian motion, b has components b _{i} as above, and the matrix field \sigma is chosen so that:(1 { 2 })/(\sigma) (x) \sigma(x) ^{\top} = a(x), \forall x \in R ^{n}

For a point

x\in 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 \tau _{D} denote the first exit time of X from D .

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

u(x) = E ^{x} \left[ g ( X _{\tau _{D}} ) ⋅ \chi _{\{ \tau _{D} < +inf ty \} } \right] + \mathbb^ \left[\int_{0}^{\tau_{D}} f(X_{t}) \, \mathrm{d} t \right]

provided that

g is a bounded function and that:E ^{x} \left[ \int ^{\tau _{D}} _{0} | f(X _{t} ) | d t \right] < + inf ty

It turns out that one further condition is required:

P ^{x} ( \tau _{D} < inf ty ) = 1 , \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) = E ^{x} \left[ g ( X _{\tau _{D}} ) \right] + E ^{x} \left[ \int ^{\tau _{D}} _{0} f(X _{t} ) d t \right]

and solves (P1) in the sense that if

l{A} denotes the characteristic operator for X (which agrees with A on C ^{2} functions), then:\begin{cases} - l{A} u(x) = f(x), & x \in D \ \displaystyle{\lim _{t \uparrow \tau _{D}} u(X _{t} )} = g ( X _{\tau _{D}} ) , & P ^{x} -a.s., \forall x \in D \end{cases} (P2)

Moreover, if $v\; \backslash in\; C^(D;\backslash mathbb)$ satisfies (P2) and there exists a constant

C such that, for all x\in D :| v(x) | \leq C \left( 1 + E ^{x} \left[ \int ^{\tau _{D}} _{0} | g(X _{s} ) | d s \right] \right)

then

v = u .

References

• 10.3792/pia/1195572706. Kakutani. Shizuo. Shizuo Kakutani. Two-dimensional Brownian motion and harmonic functions. Proc. Imp. Acad. Tokyo. 20. 10. 1944. 706 - 714. free.
• 10.3792/pia/1195572742. Kakutani. Shizuo. Shizuo Kakutani. On Brownian motions in n-space. Proc. Imp. Acad. Tokyo. 20. 9. 1944. 648 - 652. free.
• Book: Øksendal , Bernt K. . Bernt Øksendal

. Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications . Sixth. Springer. Berlin . 2003 . 3-540-04758-1. (See Section 9)