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}μ_t=b(qμ_t)-\dot{q}μ_t+sigma(qμ_t)\dot{W}_t,

qμ₀=q, dot{q}μ₀=p converges to the solution of the

equation \dot{q}_t=b(q_t)+σ(q_t)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

(Xε_t,Yε_t)$ 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).