Qualitative properties of certain piecewise deterministic Markov processes (original) (raw)
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Mathematical Biosciences and Engineering
We investigate a piecewise-deterministic Markov process, evolving on a Polish metric space, whose deterministic behaviour between random jumps is governed by some semi-flow, and any state right after the jump is attained by a randomly selected continuous transformation. It is assumed that the jumps appear at random moments, which coincide with the jump times of a Poisson process with intensity λ. The model of this type, although in a more general version, was examined in our previous papers, where we have shown, among others, that the Markov process under consideration possesses a unique invariant probability measure, say ν * λ. The aim of this paper is to prove that the map λ → ν * λ is continuous (in the topology of weak convergence of probability measures). The studied dynamical system is inspired by certain stochastic models for cell division and gene expression.
Diffusion processes on an open book and the averaging principle
Stochastic Processes and their Applications, 2004
Asymptotic problems for classical dynamical systems, stochastic processes, and PDEs can lead to stochastic processes and di erential equations on spaces with singularities. We consider the averaging principle for systems with conservation laws perturbed by small noise, where, after a change of time scale, the limiting slow motion is a di usion process on a space which is called in topology an open book: the space consisting of a number of n-dimensional manifold pieces (pages) that are glued together, sometimes several at a time, at the "binding", which is made up of manifolds of lower dimension. A di usion process on such a space is determined by di erential operators governing the process inside the pages, and gluing conditions, which determine its behavior after hitting the binding.
A new approach to the existence of invariant measures for Markovian semigroups
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2019
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we fix a convenient auxiliary measure and then we prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we obtain a unifying generalization of different versions for Harris' ergodic theorem which provides an answer to an open question of Tweedie. Also, we show that for a nonlinear SPDE on a Gelfand triple, the strict coercivity condition is sufficient to guarantee the existence of a unique invariant probability measure for the associated semigroup, once it satisfies a Harnack type inequality with power. A corollary of the main result shows that any uniformly bounded semigroup on L p possesses an invariant measure and we give some applications to sectorial perturbations of Dirichlet forms. Résumé. On établit une approche en deux étapes pour démontrer l'existence des mesure invariantes finies pour un semigroupe de Markov donné. En fixant d'abord une mesure auxiliaire convenable, on démontre ensuite des conditions équivalentes à l'existence d'une mesure invariante finie qui est absolument continue par rapport à elle. Comme applications, on obtient une généralisation unificatrice des diverses versions du théorème ergodique de Harris et on fournit une réponse à une question ouverte de Tweedie. On montre aussi que pour une EDP stochastique sur un triplet de Gelfand, la condition de coercivité stricte est suffisante pour garantir l'existence d'une seule mesure de probabilité pour le semigroupe associé, si une inégalité de type Harnack avec puissance est satisfaite. Un corollaire du résultat central montre que tout semigroupe uniformément borné sur L p possède une mesure invariante ; on donne des applications aux perturbations sectorielles des formes de Dirichlet.
Invariant measures for non-autonomous dissipative dynamical systems
Discrete and Continuous Dynamical Systems, 2014
Given a non-autonomous process U (·, ·) on a complete separable metric space X that has a pullback attractor A(·), we construct a family of invariant Borel probability measures {µt} t∈R : the measures satisfy supp µt ⊂ A(t) for all t ∈ R and the invariance property µt(E) = µτ (U (t, τ ) −1 E) for every Borel set E ∈ X. Our construction uses the generalised Banach limit. We then show that a Liouville-type equation holds for the evolution of µt under the process U (·, ·) generated by the ordinary differential equation ut = F (t, u) on a Banach space, and apply our theory to the non-autonomous 2D Navier-Stokes equations on unbounded domains satisfying a Poincaré inequality.
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Deterministic and Stochastic Difierential Inclusions with Multiple Surfaces of Discontinuity
2007
We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in Rn according to a constraint vector fleld D(¢) specifled on the boundary @G of the domain. Speciflcally, we consider equations of the form ` = µ + · + u; _ µ(t) 2 F(`(t)); a.e. t for u in an appropriate class of functions, where · is the \constraining term" in the Skorokhod problem specifled by (G;D) and F is the set- valued upper semicontinuous envelope of f. The case G = Rn (when there is no constraining mechanism) and u is absolutely continuous cor- responds to the well known setting of difierential inclusions (DI). We provide a general su-cient condition for uniqueness of solutions and Lipschitz continuity of the solution map, in the form of existence of a Lyapunov set. Here we assume (i) G is convex and admits the repre- sentation G = (iCi, where fCi;i 2 Ig is a flnite collection of disjoint, open, convex,...
Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 2001
We introduce the concept of multivalued random dynamical system (MRDS) as a measurable multivalued flow satisfying the cocycle property. We show how this is a suitable framework for the study of the asymptotic behaviour of some multivalued stochastic parabolic equations by generalizing the concept of global random attractor to the case of a MRDS. 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS
The Doeblin condition for a class of diffusions with jumps
We prove non-explosiveness and a lower bound of the spectral gap via the strong Doeblin condition for a large class of stochastic processes evolving in the inte-rior of a region D ⊆ R d with boundary ∂D according to an underlying Markov process with transition probabilities p(t, x, dy), undergoing jumps to a random point x in D with distribution ν ξ (dx) as soon as they reach a boundary point ξ. Besides usual regularity conditions on p(t, x, dy), we require a tightness condition on the family of measures ν ξ , preventing mass from escaping to the boundary. The setup can be applied to a multitude of models considered recently, including a particle system with the Bak-Sneppen dynamics from evolutionary biology.