On the study of slow-fast dynamics, when the fast process has multiple invariant measures

Motivated by applications to mathematical biology, we study the averaging problem for slow-fast systems, {\em in the case in which the fast dynamics is a stochastic process with multiple invariant measures}. We consider both the case in which the fast process is decoupled from the slow process and the case in which the two components are fully coupled. We work in the setting in which the slow process evolves according to an Ordinary Differential Equation (ODE) and the fast process is a continuous time Markov Process with finite state space and show that, in this setting, the limiting (averaged) dynamics can be described as a random ODE (that is, an ODE with random coefficients.) Keywords. Multiscale methods, Processes with multiple equilibria, Averaging, Collective Navigation, Interacting Piecewise Deterministic Markov Processes.Comment: 24 page

Poisson Equations with locally-Lipschitz coefficients and Uniform in Time Averaging for Stochastic Differential Equations via Strong Exponential Stability

We study Poisson equations and averaging for Stochastic Differential Equations (SDEs). Poisson equations are essential tools in both probability theory and partial differential equations (PDEs). Their vast range of applications includes the study of the asymptotic behaviour of solutions of parabolic PDEs, the treatment of multi-scale and homogenization problems as well as the theoretical analysis of approximations of solution of Stochastic Differential Equations (SDEs). The study of Poisson equations in non-compact state space is notoriously difficult. Results exists, but only for the case when the coefficients of the PDE are either bounded or satisfy linear growth assumptions (and even the latter case has only recently been achieved). In this paper we treat Poisson equations on non-compact state spaces for coefficients that can grow super-linearly. This is one of the two building blocks towards the second (and main) result of the paper, namely in this paper we succeed in obtaining a {\em uniform in time} (UiT) averaging result (with a rate) for SDE models with super-linearly growing coefficients. This seems to be the first UiT averaging result for slow-fast systems of SDEs (and indeed we are not aware of any other UiT multiscale theorems in the PDE or ODE literature either). Key to obtaining both our UiT averaging result and to enable dealing with the super-linear growth of the coefficients (both of the slow-fast system and of the associated Poisson equation) is conquering exponential decay in time of the space-derivatives of appropriate Markov semigroups. We refer to semigroups which enjoy this property as being Strongly Exponentially Stable.Comment: 66 pages, 2 figure

Place of Electricity in Diagnosis and Treatment of Nerve:

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