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∕, and absolute third moment bounded by M, then
|Epn(Φ) −EW(Φ)|≦ (CMV∕ρ(z,γ))n−1∕16(logn)9∕8, 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.
, and absolute third moment bounded by 