Random walks and partial differential equations.

摘要

Let A be a bounded region in 3D, and let ∂ A be its surface boundary which we assume to be absorbing. Enclose A and its boundary by a box with sides x = a, x = b, y = c, y = d, z = e, and z = f where a < b, c < d, and e < f. With delta as the step length in the random walk, assume that the intervals [a,b], [c,d], [e,f] can be subdivided into the set of points xk = a + delta k, y1 = c + delta l, and zq = e + delta q respectively with 0 ≤ k ≤ n, 0 ≤ l ≤ m, 0 ≤ q ≤ t, x n = b, ym, = d and zt = f. We say (xk,yl, z q) is an interior point of A, if it does not lie on ∂A. If one of the neighboring points lies on ∂ A or is exterior to A we call it a boundary point. In this dissertation, working primarily in 3D, we study the following problems: (i) What is the probability that a particle starting at an interior point (x, y, z) reaches the specified boundary point ( xi, yj, zv) before it reaches and is absorbed at any other boundary point?, and (ii) What is the probability that a particle starting at an interior point ( x, y, z) in the region A reaches a specified interior point (&zgr;, eta, gamma) before it reaches a boundary point and is absorbed?, and (iii) What is the "mean first passage time" for each point (x, y, z)?;Under various conditions we obtain partial differential equations of various types, and where appropriate complete the initial value problems by specifying initial or boundary conditions. These equations all result from taking limits of difference equations that serve as models for the discrete random walk problems studied.;Following the approach of S. Goldstein we also study partial difference equations for correlated random walks, leading to variants of the telegraph equation, an equation that governs the propagation of signals on telegraph lines.;Finally in the self assembling of particles, we apply the random walk concept to model, simulate, and characterize cluster growth and form.