Asymptotic Problems Related to Smoluchowski-Kramers Approximation

Asymptotic Problems Related to Smoluchowski-Kramers Approximation

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##### Date

2006-07-18

##### Authors

Chen, Zhiwei

##### Advisor

Freidlin, Mark I

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##### Abstract

According to the Smoluchowski-Kramers approximation, the solution
\qu , also referred to as ``Physical" Brownian motion, of the
Langevin's equation
$\mu\ddot{q}_{t}^{\mu,\varepsilon}=-\dot{q}_{t}^{\mu,\varepsilon}+b(q_{t}^{\mu,\varepsilon})+
\sqrt{\varepsilon}\sigma(q_{t}^{\mu,\varepsilon})\dot{W}_{t},\
q_{0}^{\mu,\varepsilon}=q,\ \dot{q}_{0}^{\mu,\varepsilon}=p$,
where $\dot{W}_t$\ is Gaussian white noise, converges to solution
of the diffusion equation
$\dot{q}_{t}^{\varepsilon}=b(\qe)+\sqrt{\varepsilon}\sigma(\qe)
\dot{W}_{t},\ q_{0}^{\varepsilon}=q$\ as $\mu\downarrow 0$\
uniformly on any finite time interval for each fixed $\ve>0$. This
is the main justification for describing the small particle motion
by a diffusion equation. However, this relation is not sufficient
for asymptotic problems when some parameter, say $\ve$, approaches
0.
We consider two asymptotic problems related to this approximation.
First, we study relations between large deviations for these
processes \qu\ and \q\ as $\ve\downarrow 0$. In particular, we
consider exit problems where relations between asymptotic exit
position, asymptotic mean exit time and some other characteristics
of the first exit of the trajectories \qu\ and \q\ from a bounded
domain are of interest. Under the framework of Freidlin-Wentzell,
these asymptotics can be represented by quasi-potential, defined
as the infimum of action functional over some set. Action
functional and quasi-potentials for \qu\ are calculated in this
paper. We establish that the asymptotics of \qu\ and \q\ are close
for small particles when $0<\mu\ll 1$. We pay special attention to
the case when $b(q)$\ is linear. Then the quasi-potentials can be
calculated explicitly and they coincide for \qu\ and \q.
Second, we study the wavefront propagation for reaction-diffusion
equations with diffusion governed by the infinitesimal generator
of process \qu\ and \q\ and reaction term governed by a nonlinear
function of KPP-type. In this case, the reaction-diffusion
equation related to the process \qu\ is degenerate in terms of
variable $(p,q)$. When the diffusion coefficient and nonlinear
term are space dependent but only changing slowly in space, we
know as $t\rightarrow \infty$, the solution of the
reaction-diffusion equation related to the process \q\ behaves
like a running wave. Characterization of the position of wavefront
for equations related to \q\ is well studied. In this work, we
identify two characterizations of the position of the wavefront
for the degenerate reaction-diffusion equation related to the
process \qu. We analyze two cases, under which we can obtain the
convergence of the wavefronts of the degenerate reaction-diffusion
equation related to \qu to those of the non-degenerate one related
to \q, for small $\mu>0$.