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- W133049696 abstract "Consider the Dirichlet problem on the planar region W consisting of the unit disc minus a sector:(W = left{ {x = left( {{x_1},{x_2}} right):0 < |x| < 1, - {a_2} < arg umentleft( x right) < {a_1}} right}) for some (0 < {a_1},{a_2} leqslant pi ,{a_1} + {a_2} >pi) (see Fig. 1). We denote by L i , i = 1,2, the line segment (left{ {operatorname{x} :argumentleft( x right) = {{left( { - 1} right)}^{i - 1}}{a_i},0 leqslant |x| leqslant 1} right}) in ∂W. Let ϕ be a piecewise continuous function on ∂W which vanishes on L 1 U L 2 and let u(ϕ,•) be the harmonic function on W with boundary values ϕ. Thus △ϕ = O. The discrete analogue, or the analogue for a simple random walk with step size h, is a function u h (ϕ, •) on hℤ 2 which satisfies $${Delta _h}{mu _h}left( {varphi ,chi } right): = frac{1}{4}sumlimits_{i = 1}^2 {left[ {{mu _h}left( {varphi ,chi + {h_{ei}}} right) + {mu _h}left( {varphi ,chi - {h_{ei}}} right)} right]} - {mu _h}left( {varphi ,chi } right) = 0$$ for x ∈ hℤ2 ∩ W, where e i is the i — th coordinate vector. Moreover u h (ϕ, •) should be “close to ϕ on the discrete boundary of W”. We prove an asymptotic relation (as h ↓ 0) between u h ϕ, hv) and u(ϕ,hv) for fixed υ ∈ ℤ2. This can be used to find the asymptotic behavior (as n → ∞) of the probability of a simple random walk “starting at ∞” to hit the “triangular” region (left{ {x in {Z^2}:|x| leqslant n, - {alpha _2} leqslant operatorname{argument} left( x right) leqslant {alpha _1}} right}) near the origin. This kind of estimate is useful for a modified version of Diffusion—Limited Aggregation.Key Words and PhrasesDirichlet problemBrownian motionrandom walkdiffusion-limited aggregationhitting probabilities" @default.
- W133049696 created "2016-06-24" @default.
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- W133049696 date "1991-01-01" @default.
- W133049696 modified "2023-09-25" @default.
- W133049696 title "Relations Between Solutions to a Discrete and Continuous Dirichlet Problem" @default.
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- W133049696 doi "https://doi.org/10.1007/978-1-4612-0459-6_17" @default.
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