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

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2009##### Author

Spiliopoulos, Konstantinos

##### Advisor

Freidlin, Mark I

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Show full item record##### 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}<super>μ</super><sub>t</sub>=b(q<super>μ</super><sub>t</sub>)-\dot{q}<super>μ</super><sub>t</sub>+sigma(q<super>μ</super><sub>t</sub>)\dot{W}<sub>t</sub>,
q<super>μ</super><sub>0</sub>=q, dot{q}<super>μ</super><sub>0</sub>=p converges to the solution of the
equation \dot{q}<sub>t</sub>=b(q<sub>t</sub>)+σ(q<sub>t</sub>)dot{W}<sub>t</sub>, q<sub>0</sub>=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 <bold>R</bold><sub>+</sub><super>n</super>={(x<super>1</super>,...,x<super>n</super>)
in <bold>R</bold><super>n</super>: x<super>1</super>= 0}. After proving that such a process
exists and is well defined, we prove that the Langevin process with
reflection at x<super>1</super>=0 converges in distribution to the diffusion
process with reflection on the boundary of <bold>R</bold><sub>+</sub><super>n</super>. 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 D<super>ε</super>, for
linear second order differential equations with nonlinear boundary
conditions. The underlying stochastic process is the Wiener process
(X<super>ε</super><sub>t</sub>,Y<super>ε</super><sub>t</sub>)$ in the narrow domain
D<super>ε</super> 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).

University of Maryland, College Park, MD 20742-7011 (301)314-1328.

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