1,189 research outputs found
Torsional rigidity for cylinders with a Brownian fracture
We obtain bounds for the expected loss of torsional rigidity of a cylinder
of length due to a Brownian
fracture that starts at a random point in and runs until the first
time it exits . These bounds are expressed in terms of the geometry
of the cross-section . It is shown that if is a
disc with radius , then in the limit as the expected
loss of torsional rigidity equals for some . We derive
bounds for in terms of the expected Newtonian capacity of the trace of a
Brownian path that starts at the centre of a ball in with radius
and runs until the first time it exits this ball.Comment: 18 page
Intermittency in a catalytic random medium
In this paper, we study intermittency for the parabolic Anderson equation
, where , is the diffusion constant, is the
discrete Laplacian and is a
space-time random medium. We focus on the case where is times
the random medium that is obtained by running independent simple random walks
with diffusion constant starting from a Poisson random field with
intensity . Throughout the paper, we assume that
. The solution of the equation describes
the evolution of a ``reactant'' under the influence of a ``catalyst''
. We consider the annealed Lyapunov exponents, that is, the exponential
growth rates of the successive moments of , and show that they display an
interesting dependence on the dimension and on the parameters
, with qualitatively different intermittency behavior
in , in and in . Special attention is given to the
asymptotics of these Lyapunov exponents for and .Comment: Published at http://dx.doi.org/10.1214/009117906000000467 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Stretched Exponential Relaxation in the Biased Random Voter Model
We study the relaxation properties of the voter model with i.i.d. random
bias. We prove under mild condions that the disorder-averaged relaxation of
this biased random voter model is faster than a stretched exponential with
exponent , where depends on the transition rates
of the non-biased voter model. Under an additional assumption, we show that the
above upper bound is optimal. The main ingredient of our proof is a result of
Donsker and Varadhan (1979).Comment: 14 pages, AMS-LaTe
Heat content and inradius for regions with a Brownian boundary
In this paper we consider , Brownian motion of time length , in -dimensional Euclidean space and on the -dimensional
torus . We compute the expectation of (i) the heat content at time
of for fixed and in the
limit , when is kept at temperature 1 for all and has initial temperature 0, and (ii)
the inradius of for in the
limit .Comment: 13 page
Intermittency on catalysts: three-dimensional simple symmetric exclusion
We continue our study of intermittency for the parabolic Anderson model
in a space-time random medium
, where is a positive diffusion constant, is the lattice
Laplacian on , , and is a simple symmetric exclusion
process on in Bernoulli equilibrium. This model describes the evolution
of a \emph{reactant} under the influence of a \emph{catalyst} .
In G\"artner, den Hollander and Maillard (2007) we investigated the behavior
of the annealed Lyapunov exponents, i.e., the exponential growth rates as
of the successive moments of the solution . This led to an
almost complete picture of intermittency as a function of and . In
the present paper we finish our study by focussing on the asymptotics of the
Lyaponov exponents as in the \emph{critical} dimension ,
which was left open in G\"artner, den Hollander and Maillard (2007) and which
is the most challenging. We show that, interestingly, this asymptotics is
characterized not only by a \emph{Green} term, as in , but also by a
\emph{polaron} term. The presence of the latter implies intermittency of
\emph{all} orders above a finite threshold for .Comment: 38 page
Intermittency on catalysts
The present paper provides an overview of results obtained in four recent
papers by the authors. These papers address the problem of intermittency for
the Parabolic Anderson Model in a \emph{time-dependent random medium},
describing the evolution of a ``reactant'' in the presence of a ``catalyst''.
Three examples of catalysts are considered: (1) independent simple random
walks; (2) symmetric exclusion process; (3) symmetric voter model. The focus is
on the annealed Lyapunov exponents, i.e., the exponential growth rates of the
successive moments of the reactant. It turns out that these exponents exhibit
an interesting dependence on the dimension and on the diffusion constant.Comment: 11 pages, invited paper to appear in a Festschrift in honour of
Heinrich von Weizs\"acker, on the occasion of his 60th birthday, to be
published by Cambridge University Pres
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