We study the continuous-time limit of a class of Markov chains coming from the evolution of classical open systems undergoing repeated interactions. This repeated interaction model has been initially developed for dissipative quantum systems in Attal and Pautrat (2006) and was recently set up...

We prove the Local Asymptotic Mixed Normality property from high frequency observations, of a continuous time process solution of a stochastic differential equation driven by a pure jump Lévy process. The process is observed on the fixed time interval [0,1] and the parameter appears in the...

We consider a backward stochastic differential equation in a Markovian framework for the pair of processes (Y,Z), with generator with quadratic growth with respect to Z. Under non-degeneracy assumptions, we prove an analogue of the well-known Bismut–Elworthy formula when the generator has...

Let X1,X2,… be a sequence of i.i.d. random variables, with mean zero and variance one and let Sn=(X1+⋯+Xn)/n. An old and celebrated result of Prohorov (1952) asserts that Sn converges in total variation to the standard Gaussian distribution if and only if Sn0 has an absolutely continuous...

This paper deals with homogenization of divergence form second order parabolic operators whose coefficients are periodic with respect to the spatial variables and random stationary in time. Under proper mixing assumptions, we study the limit behaviour of the normalized difference between...

We investigate branching properties of the solution of a fragmentation equation for the mass distribution and we properly associate a continuous time càdlàg Markov process on the space S↓ of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel induces a specific...

We obtain limit theorems for a class of nonlinear discrete-time processes X(n) called the kth order Volterra processes of order k. These are moving average kth order polynomial forms: X(n)=∑0i1,…,ik∞a(i1,…,ik)ϵn−i1…ϵn−ik, where {ϵi} is i.i.d. with Eϵi=0, Eϵi2=1, where a(⋅)...

We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power p, for p∈[0,2). The asymptotic behaviour of the right-most particle for this system is already known; in...

We provide an example of a discrete-time Markov process on the three-dimensional infinite integer lattice with Zq-invariant Bernoulli-increments which has as local state space the cyclic group Zq. We show that the system has a unique invariant measure, but remarkably possesses an invariant set...

Consider a centred random walk in dimension one with a positive finite variance σ2, and let τB be the hitting time for a bounded Borel set B with a non-empty interior. We prove the asymptotic Px(τBn)∼2/πσ−1VB(x)n−1/2 and provide an explicit formula for the limit VB as a function of...