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This is a survey paper about reciprocal processes. The bridges of a Markov process are also Markov. But an arbitrary mixture of these bridges fails to be Markov in general. However, it still enjoys the interesting properties of a reciprocal…
For both continuous-time and discrete-time Markov Chains, we provide criteria for inverse problems of classical types of ergodicity: (ordinary) erogodicity, algebraic ergodicity, exponential ergodicity and strong ergodicity. Our criteria…
We introduce Markov substitute processes, a new model at the crossroad of statistics and formal grammars, and prove its main property : Markov substitute processes with a given support form an exponential family.
Ergodic properties of rational maps are studied, generalising the work of F.\ Ledrappier. A new construction allows for simpler proofs of stronger results. Very general conformal measures are considered. Equivalent conditions are given for…
In this paper, we introduce the notion of Bi-entangled hidden Markov processes. These are hidden quantum processes where the hidden processes themselves exhibit entangled Markov process, and the observable processes also exhibit…
Recently O. Sarig has introduced and explored the concept of positively recurrent functions. In this paper we construct a natural wide class of such functions and we showthat they have stronger ergodic properties than the general functions…
We discuss a Markov jump process regarded as a variant of the CIR (Cox-Ingersoll-Ross) model and its infinite-dimensional extension. These models belong to a class of measure-valued branching processes with immigration, whose jump…
We establish general theorems quantifying the notion of recurrence --- through an estimation of the moments of passage times --- for irreducible continuous-time Markov chains on countably infinite state spaces. Sharp conditions of…
We study ergodic properties of some Markov chains models in random environments when the random Markov kernels that define the dynamic satisfy some usual drift and small set conditions but with random coefficients. In particular, we adapt a…
We consider a type of Markov property for set-indexed processes which is satisfied by all processes with independent increments and which allows us to introduce a transition system theory leading to the construction of the process. A…
Spatial birth-and-death processes with a finite number of particles are obtained as unique solutions to certain stochastic equations. Conditions are given for existence and uniqueness of such solutions, as well as for continuous dependence…
This paper focuses on optimizing probabilities of events of interest defined over general controlled discrete-time Markov processes. It is shown that the optimization over a wide class of $\omega$-regular properties can be reduced to the…
This work develops asymptotic properties of a class of switching jump diffusion processes. The processes under consideration may be viewed as a number of jump diffusion processes modulated by a random switching mechanism. The underlying…
The first motivation of this paper is to study stationarity and ergodic properties for a general class of time series models defined conditional on an exogenous covariates process. The dynamic of these models is given by an autoregressive…
Improved rates of convergence for ergodic homogeneous Markov chains are studied. In comparison to the earlier papers the setting is also generalised to the case without a unique dominated measure. Examples are provided where the new bound…
We study the relationship between two classical approaches for quantitative ergodic properties : the first one based on Lyapunov type controls and popularized by Meyn and Tweedie, the second one based on functional inequalities (of…
Recently Belopolskaya and Suhov studied Markov process's in a random environment, where the environment changes in likeness to a Markov process. Constructions were made to allow the process to "interact with the environment", this was done…
We consider the down/up crossing property of weighted Markov branching processes. The joint probability distribution of multi crossing numbers of such processes are obtained. In particular, for Markov branching processes, the probability…
This note provides several recent progresses in the study of long time behavior of Markov processes. The examples presented below are related to other scientific fields as PDE's, physics or biology. The involved mathematical tools as…
We discuss multiple versions of rational ergodicity and rational weak mixing for "nice" transformations, including Markov shifts, certain interval maps and hyperbolic geodesic flows. These properties entail multiple recurrence.