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- W2523063908 abstract "The probability p(x) that Brownian motion with drift, starting at x, hits an obstacle is analyzed. The obstacle Q is a compact subset of R/. It is shown that p(x) is expressible in terms of the field U(x) scattered by Q when it is hit by a plane wave. Therefore results for U(x), and methods for finding U(x), can be used to determine p(x). We illustrate this by obtaining exact and asymptotic results for p(x) when Q is a slit in R2, and asymptotic results when Q is a disc in R3. photon absorption by ions, and of other diffusion processes. We shall show that p(x) is proportional to the field U(x) scattered by Q when it is hit by a plane wave with the imaginary propagation constant k = -iv/2D, where U satisfies the Helmholtz equation and the total field vanishes on 2Q. Therefore one can find p by using results and methods developed for calculating U. We shall illustrate this relation by finding p(x) when Q is a slit of length 2L along the y-axis of R2 and the drift v is in the positive x direction. This is the model of chemotaxis introduced by Landau et al. (1) to describe how a cell determines which way to move toward the source of a chemical attractant. They also considered p+ (x) and p_ (x), the probabilities that Brownian motion hits the right and left sides of the slit, respectively, which we shall calculate also. Scattering by an infinite strip of width 2L, which intersects R2 in a slit of length 2L, has been studied in great detail, and many of the results for U(x) are given in Bowman, Senior, and Uslenghi (2, Chapter 4). From them, we shall see that the main results of (1) can be obtained by using the relation between p(x) and U(x). In fact, much more detailed asymptotic results can be found both for kL large and for kL" @default.
- W2523063908 created "2016-09-30" @default.
- W2523063908 creator A5069435472 @default.
- W2523063908 date "2000-01-01" @default.
- W2523063908 modified "2023-09-27" @default.
- W2523063908 title "1. Introduction. Consider a particle starting at the point x and executing a Brownian motion with diffusion coefficient D and drift velocity v in the presence of an absorbing obstacle Q. The probability p(x) that the particle hits Q in a finite time" @default.
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