Introduction: Kakutani's Solution To The Classical Dirichlet Problem
Let D be a domain (an open and connected set) in Rn. Let Δ be the Laplace operator, let g be a bounded function on the boundary ∂D, and consider the problem
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.
Read more about this topic: Stochastic Processes And Boundary Value Problems
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