153 research outputs found
Mixed-mode oscillations and interspike interval statistics in the stochastic FitzHugh-Nagumo model
We study the stochastic FitzHugh-Nagumo equations, modelling the dynamics of
neuronal action potentials, in parameter regimes characterised by mixed-mode
oscillations. The interspike time interval is related to the random number of
small-amplitude oscillations separating consecutive spikes. We prove that this
number has an asymptotically geometric distribution, whose parameter is related
to the principal eigenvalue of a substochastic Markov chain. We provide
rigorous bounds on this eigenvalue in the small-noise regime, and derive an
approximation of its dependence on the system's parameters for a large range of
noise intensities. This yields a precise description of the probability
distribution of observed mixed-mode patterns and interspike intervals.Comment: 36 page
Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise
We study stochastic partial differential equations of the reaction-diffusion
type. We show that, even if the forcing is very degenerate (i.e. has not full
rank), one has exponential convergence towards the invariant measure. The
convergence takes place in the topology induced by a weighted variation norm
and uses a kind of (uniform) Doeblin condition.Comment: 10 pages, 1 figur
Convergence to equilibrium for many particle systems
The goal of this paper is to give a short review of recent results of the
authors concerning classical Hamiltonian many particle systems. We hope that
these results support the new possible formulation of Boltzmann's ergodicity
hypothesis which sounds as follows. For almost all potentials, the minimal
contact with external world, through only one particle of , is sufficient
for ergodicity. But only if this contact has no memory. Also new results for
quantum case are presented
Minimizing or Maximizing the Expected Time to Reach Zero
1 online resource (PDF, 26 pages
On infinite-volume mixing
In the context of the long-standing issue of mixing in infinite ergodic
theory, we introduce the idea of mixing for observables possessing an
infinite-volume average. The idea is borrowed from statistical mechanics and
appears to be relevant, at least for extended systems with a direct physical
interpretation. We discuss the pros and cons of a few mathematical definitions
that can be devised, testing them on a prototypical class of infinite
measure-preserving dynamical systems, namely, the random walks.Comment: 34 pages, final version accepted by Communications in Mathematical
Physics (some changes in Sect. 3 -- Prop. 3.1 in previous version was
partially incorrect
A lower lipschitz condition for the stable subordinator
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47646/1/440_2004_Article_BF00538471.pd
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