Vol. 42, No. 3, 1972

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A probabilistic method for the rate of convergence to the Dirichlet problem

David F. Fraser

Vol. 42 (1972), No. 3, 657–665

The expectation Ep(Φ) approximates the solution u(z) = EW(Φ) of the Dirichlet problem for a plane domain D with boundary conditions ϕ on the boundary γ of D, where W is Wiener measure, P is the measure generated by a random walk which approximates Brownian motion beginning at z, and Φ is the functional on paths which equals the value of ϕ at the point where the path first meets γ. This paper develops a specific rate of convergence. If γ is C2, and Pn is generated by random walks beginning at z, with independent increments in the coordinate directions at intervals 1∕n, with mean zero, variance 1√n--, and absolute third moment bounded by M, then |Epn(Φ) EW(Φ)|(CMV∕ρ(z,γ))n116(log n)98, where V is the total variation of ϕ on γ,ρ(z,γ) is the distance from z to γ, and C is a constant depending only on γ.

Mathematical Subject Classification 2000
Primary: 60J15
Secondary: 31A25, 65N15
Received: 15 July 1969
Revised: 15 May 1972
Published: 1 September 1972
David F. Fraser