97 research outputs found
Ergodic properties of a model for turbulent dispersion of inertial particles
We study a simple stochastic differential equation that models the dispersion
of close heavy particles moving in a turbulent flow. In one and two dimensions,
the model is closely related to the one-dimensional stationary Schroedinger
equation in a random delta-correlated potential. The ergodic properties of the
dispersion process are investigated by proving that its generator is
hypoelliptic and using control theory
Analysis of equilibrium states of Markov solutions to the 3D Navier-Stokes equations driven by additive noise
We prove that every Markov solution to the three dimensional Navier-Stokes
equation with periodic boundary conditions driven by additive Gaussian noise is
uniquely ergodic. The convergence to the (unique) invariant measure is
exponentially fast.
Moreover, we give a well-posedness criterion for the equations in terms of
invariant measures. We also analyse the energy balance and identify the term
which ensures equality in the balance.Comment: 32 page
Persistence for stochastic difference equations: A mini-review
Understanding under what conditions populations, whether they be plants,
animals, or viral particles, persist is an issue of theoretical and practical
importance in population biology. Both biotic interactions and environmental
fluctuations are key factors that can facilitate or disrupt persistence. One
approach to examining the interplay between these deterministic and stochastic
forces is the construction and analysis of stochastic difference equations
where represents the state of the
populations and is a sequence of random variables
representing environmental stochasticity. In the analysis of these stochastic
models, many theoretical population biologists are interested in whether the
models are bounded and persistent. Here, boundedness asserts that
asymptotically tends to remain in compact sets. In contrast, persistence
requires that tends to be "repelled" by some "extinction set" . Here, results on both of these proprieties are reviewed for single
species, multiple species, and structured population models. The results are
illustrated with applications to stochastic versions of the Hassell and Ricker
single species models, Ricker, Beverton-Holt, lottery models of competition,
and lottery models of rock-paper-scissor games. A variety of conjectures and
suggestions for future research are presented.Comment: Accepted for publication in the Journal of Difference Equations and
Application
Optimal scaling of average queue sizes in an input-queued switch: an open problem
We review some known results and state a few versions of an open problem related to the scaling of the total queue size (in steady state) in an n×n input-queued switch, as a function of the port number n and the load factor ρ. Loosely speaking, the question is whether the total number of packets in queue, under either the maximum weight policy or under an optimal policy, scales (ignoring any logarithmic factors) as O(n/(1 − ρ)).National Science Foundation (U.S.) (Grant CCF-0728554
Billiards in a general domain with random reflections
We study stochastic billiards on general tables: a particle moves according
to its constant velocity inside some domain until it hits the boundary and bounces randomly inside according to some
reflection law. We assume that the boundary of the domain is locally Lipschitz
and almost everywhere continuously differentiable. The angle of the outgoing
velocity with the inner normal vector has a specified, absolutely continuous
density. We construct the discrete time and the continuous time processes
recording the sequence of hitting points on the boundary and the pair
location/velocity. We mainly focus on the case of bounded domains. Then, we
prove exponential ergodicity of these two Markov processes, we study their
invariant distribution and their normal (Gaussian) fluctuations. Of particular
interest is the case of the cosine reflection law: the stationary distributions
for the two processes are uniform in this case, the discrete time chain is
reversible though the continuous time process is quasi-reversible. Also in this
case, we give a natural construction of a chord "picked at random" in
, and we study the angle of intersection of the process with a
-dimensional manifold contained in .Comment: 50 pages, 10 figures; To appear in: Archive for Rational Mechanics
and Analysis; corrected Theorem 2.8 (induced chords in nonconvex subdomains
Stationary distributions for diffusions with inert drift
Consider reflecting Brownian motion in a bounded domain in that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential
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