Letting the initial condition of a PDE be random is interesting when considering complex phenomena. For 2D-Navier-Stokes equations, it is for instance an attempt to take into account the turbulence arising with high velocities and low viscosities. The solutions of these PDEs are random and their...

We study a Linear-Quadratic Regulation (LQR) problem with Lévy processes and establish the closeness property of the solution of the multi-dimensional Backward Stochastic Riccati Differential Equation (BSRDE) with Lévy processes. In particular, we consider multi-dimensional and one-dimensional...

We prove a functional central limit theorem for the position of a tagged particle in the one-dimensional asymmetric simple exclusion process for hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the...

We develop a notion of nonlinear expectation-G-expectation-generated by a nonlinear heat equation with infinitesimal generator G. We first study multi-dimensional G-normal distributions. With this nonlinear distribution we can introduce our G-expectation under which the canonical process is a...

Let x(s), s[set membership, variant]Rd be a Gaussian self-similar random process of index H. We consider the problem of log-asymptotics for the probability pT that x(s), x(0)=0 does not exceed a fixed level in a star-shaped expanding domain T[dot operator][Delta] as T--[infinity]. We solve the...

Many real-life time series exhibit clusters of outlying observations that cannot be adequately modeled by a Gaussian distribution. Heavy-tailed distributions such as the Pareto distribution have proved useful in modeling a wide range of bursty phenomena that occur in areas as diverse as finance,...

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G....

We prove functional limits theorems for the occupation time process of a system of particles moving independently in according to a symmetric [alpha]-stable Lévy process, and starting from an inhomogeneous Poisson point measure with intensity measure , and other related measures. In contrast to...

We study the random walk in a random environment on , where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) a random walk in a random environment perturbed from Sinai's regime; (ii) a simple random walk with a random...

We investigate the cumulative scenery process associated with random walks in independent, identically distributed random sceneries under the assumption that the scenery variables satisfy Cramér's condition. We prove moderate deviation principles in dimensions d=2, covering all those regimes...