We consider the spatial $𝛬$-Fleming–Viot process model (Electron. J. Probab.15(2010) 162–216) for frequencies of genetic types in a population living in ${ℝ}^d$, in the special case in which there are just two types of individuals, labelled $0$ and $1$. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the dynamics...

Let $Y$ be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process $X:\mathrm{d}{Y}_t=a\left(X_t\right)Y_t\mathrm{d}t+\sigma \left(X_t\right)\mathrm{d}{W}_t,Y_0=y_0$. We establish that under the condition $\alpha =E_{\mu }\left(a\left(X_0\right)\right)\<0$ with $\mu$ the stationary distribution of the regime process $X$, the diffusion $Y$ is ergodic. We also consider conditions for the existence of moments for the invariant law of $Y$ when $X$ is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that conditionally to $X$, $Y$ is gaussian on the other hand, we give...

Let Y be a Ornstein–Uhlenbeck diffusion governed by a stationary and ergodic process X : dYt = a(Xt)Ytdt + σ(Xt)dWt,Y0 = y0. We establish that under the condition α = Eµ(a(X0)) < 0 with μ the stationary distribution of the regime process X, the diffusion Y is ergodic. We also consider conditions for the existence of moments for the invariant law of Y when X is a Markov jump process having a finite number of states. Using results on random difference equations on one hand and the fact that...

Let ${{\mu }_t}_{t>0}$ be a symmetric semigroup of stable measures on a homogeneous group, with smooth Lévy measure. Applying Malliavin calculus for jump processes we prove that the measures ${\mu }_t$ have smooth densities.

We consider the Fluctuation Dissipation Theorem (FDT) of statistical physics from a mathematical perspective. We formalize the concept of “linear response function” in the general framework of Markov processes. We show that for processes out of equilibrium it depends not only on the given Markov process X(s) but also on the chosen perturbation of it. We characterize the set of all possible response functions for a given Markov process and show that at equilibrium they all satisfy the FDT. That is,...

We consider a dynamical system in $ℝ$ driven by a vector field -U', where U is a multi-well potential satisfying some regularity conditions. We perturb this dynamical system by a Lévy noise of small intensity and such that the heaviest tail of its Lévy measure is regularly varying. We show that the perturbed dynamical system exhibits metastable behaviour i.e. on a proper time scale it reminds of a Markov jump process taking values in the local minima of the potential U. Due to the heavy-tail nature...

We study minimal thinness in the half-space $H:=\{x=(\tilde{x},x_d):\tilde{x}\in {ℝ}^{d-1},x_d\&gt;0\}$ for a large class of subordinate Brownian motions. We show that the same test for the minimal thinness of a subset of $H$ below the graph of a nonnegative Lipschitz function is valid for all processes in the considered class. In the classical case of Brownian motion this test was proved by Burdzy.

We introduce and investigate a new model of a finite number of particles jumping forward on the real line. The jump lengths are independent of everything, but the jump rate of each particle depends on the relative position of the particle compared to the center of mass of the system. The rates are higher for those left behind, and lower for those ahead of the center of mass, providing an attractive interaction keeping the particles together. We prove that in the fluid limit, as the number of particles...

In [A. Genadot and M. Thieullen, Averaging for a fully coupled piecewise-deterministic markov process in infinite dimensions. Adv. Appl. Probab. 44 (2012) 749–773], the authors addressed the question of averaging for a slow-fast Piecewise Deterministic Markov Process (PDMP) in infinite dimensions. In the present paper, we carry on and complete this work by the mathematical analysis of the fluctuations of the slow-fast system around the averaged limit. A central limit theorem is derived and the associated...

This paper deals with nonlinear filtering problems with delays, i.e., we consider a system (X,Y), which can be represented by means of a system (X,Ŷ), in the sense that Yt = Ŷa(t), where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on the existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory....

In this paper the adaptive control problem for a continuous infinite time-varying stochastic control system with jumps in parameters and quadratic cost is investigated. It is assumed that the unknown coefficients of the system have limits as time tends to infinity and the boundary system is absolutely observable and stabilizable. Under these assumptions it is shown that the optimal value of the quadratic cost can be reached based only on the values of these limits, which, in turn, can be estimated...