Asymptotic problems for stochastic processes with reflection and related PDE's

Abstract

Asymptotic problems for stochastic processes with reflection and for related partial differential equations (PDE's) are considered in this thesis. The stochastic processes that we study, depend on a small parameter and are restricted to move in the interior of some domain, while having instantaneous reflection at the boundary of the domain. These stochastic processes are closely related to corresponding PDE problems that depend on a small parameter. We are interested in the behavior of these stochastic processes and of the solutions to the corresponding PDE problems as this small parameter goes to zero. In particular, we consider two problems that are related to stochastic processes with reflection at the boundary of some domain. Firstly, we study the Smoluchowski-Kramers approximation for the Langevin equation with reflection. According to the Smoluchowski-Kramers approximation, the solution of the equation μ\ddot{q}μt=b(qμt)-\dot{q}μt+sigma(qμt)\dot{W}t, qμ0=q, dot{q}μ0=p converges to the solution of the equation \dot{q}t=b(qt)+σ(qt)dot{W}t, q0=q as μ← 0. We consider here a similar result for the Langevin process with elastic reflection on the boundary of the half space, i.e. on partial R+n={(x1,...,xn) in Rn: x1= 0}. After proving that such a process exists and is well defined, we prove that the Langevin process with reflection at x1=0 converges in distribution to the diffusion process with reflection on the boundary of R+n. This convergence is the main justification for using a first order equation, instead of a second order one, to describe the motion of a small mass particle that is restricted to move in the interior of some domain and reflects elastically on its boundary. Secondly, we study the second initial boundary problem in a narrow domain of width ε<< 1, denoted by Dε, for linear second order differential equations with nonlinear boundary conditions. The underlying stochastic process is the Wiener process (Xεt,Yεt)$ in the narrow domain Dε with instantaneous normal reflection at its boundary. Using probabilistic methods we show that the solution of such a problem converges to the solution of a standard reaction-diffusion equation in a domain of reduced dimension as &epsilon→0. This reduction allows to obtain some results concerning wave front propagation in narrow domains. In particular, we describe conditions leading to jumps of the wave front. This problem is important in applications (e.g., thin waveguides).