- Stochastic processes and boundary value problems
In
mathematics , someboundary value problem s can be solved using the methods of stochastic analysis. Perhaps the most celebrated example isShizuo Kakutani 's 1944 solution of theDirichlet problem for theLaplace operator usingBrownian motion . However, it turns out that for a large class of semi-elliptic second-orderpartial differential equations the associated Dirichlet boundary value problem can be solved using anItō process that solves an associatedstochastic differential equation .Introduction: Kakutani's solution to the classical Dirichlet problem
Let "D" be a domain (an open and connected set) in R"n". Let Δ be the
Laplace operator , let "g" be abounded function on the boundary ∂"D", and consider the problem:egin{cases} - Delta u(x) = 0, & x in D; \ displaystyle{lim_{y o x} 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 the (random) first exit time from "D" for a canonicalBrownian motion starting at "x".The Dirichlet-Poisson problem
Let "D" be a domain in R"n" and let "L" be a semi-elliptic differential operator on "C"2(R"n"; R) of the form
: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 "b""i" and "a""ij" are
continuous function s and all theeigenvalue s of the matrix "a"("x") = ("a""ij"("x")) are non-negative. Let "f" ∈ "C"("D"; R) and "g" ∈ "C"(∂"D"; R). Consider the Poisson problem:egin{cases} - L u(x) = f(x), & x in D; \ displaystyle{lim_{y o 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 "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:mathrm{d} X_{t} = b(X_{t}) , mathrm{d} t + sigma (X_{t}) , mathrm{d} B_{t},
where "B" is "n"-dimensional Brownian motion, "b" has components "b""i" as above, and the matrix field "σ" is chosen so that
:frac1{2} sigma (x) sigma(x)^{ op} = a(x) mbox{ for all } x in mathbf{R}^{n}.
For a point "x" ∈ 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 "τ""D" denote the first exit time of "X" from "D".
In this notation, the candidate solution for (P1) is
:u(x) = mathbf{E}^{x} left [ g ig( X_{ au_{D ig) cdot chi_{{ au_{D} < + infty ight] + mathbf{E}^{X} left [ int_{0}^{ au_{D f(X_{t}) , mathrm{d} t ight]
provided that "g" is a
bounded function and that:mathbf{E}^{x} left [ int_{0}^{ au_{D ig| f(X_{t}) ig| , mathrm{d} t ight] < + infty.
It turns out that one further condition is required:
:mathbf{P}^{x} ig [ au_{D} < + infty ig] = 1 mbox{ for all } x in D,
i.e., 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) = mathbf{E}^{x} left [ g ig( X_{ au_{D ig) ight] + mathbf{E}^{x} left [ int_{0}^{ au_{D f(X_{t}) , mathrm{d} t ight]
and solves (P1) in the sense that if mathcal{A} denotes the characteristic operator for "X" (which agrees with "A" on "C"2 functions), then
:egin{cases} - mathcal{A} u(x) = f(x), & x in D; \ displaystyle{lim_{t uparrow au_{D u(X_{t})} = g ig( X_{ au_{D ig), & mathbf{P}^{x} mbox{-a.s., for all } x in D. end{cases} quad mbox{(P2)}
Moreover, if "v" ∈ "C"2("D"; R) satisfies (P2) and there exists a constant "C" such that, for all "x" ∈ "D",
:v(x) | leq C left( 1 + mathbf{E}^{x} left [ int_{0}^{ au_{D ig| g(X_{s}) ig| , mathrm{d} s ight] ight),
then "v" = "u".
References
* cite journal
last = Kakutani
first = Shizuo
authorlink= Shizuo Kakutani
title = Two-dimensional Brownian motion and harmonic functions
journal = Proc. Imp. Acad. Tokyo
volume = 20
year = 1944
pages = 706–714
* cite journal
last = Kakutani
first = Shizuo
authorlink= Shizuo Kakutani
title = On Brownian motions in "n"-space
journal = Proc. Imp. Acad. Tokyo
volume = 20
year = 1944
pages = 648–652
* cite book
last = Øksendal
first = Bernt K.
authorlink = Bernt Øksendal
title = Stochastic Differential Equations: An Introduction with Applications
edition = Sixth edition
publisher=Springer
location = Berlin
year = 2003
isbn = 3-540-04758-1 (See Section 9)
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