The Fokker—Planck Equation in Several Dimensions: the Asymptotic Exit Problem

Abstract

In this chapter we analyse the problem of exit from a domain Ω in ℝⁿ. We will only deal systematically with the problem that the boundary ∂Ω is absorbing. The situation that a part of the boundary is reflecting or not attainable will arise in some special problems. The case that both the drift and normal component of the diffusion vanish at the boundary will be examined in Chap. 7. The latter is of special importance in the field of stochastic dynamics of biological populations (expected extinction time). In this chapter it is assumed that the forward and backward operator are uniformly elliptic. For other approaches to this type of Fokker—Planck problems we refer to Graham and Tel (1985), Graham et al. (1985), Knessl et al. (1985), Hagan et al. (1989), Day (1990), Talkner (1987), Maier and Stein (1993), Lythe (1995) and Khasminskii and Yin (1996ab).