82 research outputs found
Small mass asymptotic for the motion with vanishing friction
We consider the small mass asymptotic (Smoluchowski-Kramers approximation)
for the Langevin equation with a variable friction coefficient. The friction
coefficient is assumed to be vanishing within certain region. We introduce a
regularization for this problem and study the limiting motion for the
1-dimensional case and a multidimensional model problem. The limiting motion is
a Markov process on a projected space. We specify the generator and boundary
condition of this limiting Markov process and prove the convergence.Comment: final version for publication, accepted by Stochastic Processes and
their Application
On diffusion in narrow random channels
We consider in this paper a solvable model for the motion of molecular
motors. Based on the averaging principle, we reduce the problem to a diffusion
process on a graph. We then calculate the effective speed of transportation of
these motors.Comment: 23 pages, 3 figures, comments are welcom
On second order elliptic equations with a small parameter
The Neumann problem with a small parameter
is
considered in this paper. The operators and are self-adjoint second
order operators. We assume that has a non-negative characteristic form
and is strictly elliptic. The reflection is with respect to inward
co-normal unit vector . The behavior of
is effectively described via
the solution of an ordinary differential equation on a tree. We calculate the
differential operators inside the edges of this tree and the gluing condition
at the root. Our approach is based on an analysis of the corresponding
diffusion processes.Comment: 28 pages, 1 figure, revised versio
On the Smoluchowski-Kramers approximation for SPDEs and its interplay with large deviations and long time behavior
We discuss here the validity of the small mass limit (the so-called
Smoluchowski-Kramers approximation) on a fixed time interval for a class of
semi-linear stochastic wave equations, both in the case of the presence of a
constant friction term and in the case of the presence of a constant magnetic
field. We also consider the small mass limit in an infinite time interval and
we see how the approximation is stable in terms of the invariant measure and of
the large deviation estimates and the exit problem from a bounded domain of the
space of square integrable functions.Comment: arXiv admin note: text overlap with arXiv:1403.574
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