Abstract: We consider a class of nonlinear population models on a two-dimensional lattice which are influenced by a small random potential, and we show that on large temporal and spatial scales the population density is well described by the continuous parabolic Anderson model, a linear but singular stochastic PDE. The proof is based on a discrete formulation of paracontrolled distributions on unbounded lattices which is of independent interest because it can be applied to prove the convergence of a wide range of lattice models. This is joint work with Jörg Martin.
SEMINARIO DI PROBABILITA', ANALISI STOCASTICA E STATISTICA
Chemical reaction networks are mathematical models used in
biochemistry, as well as in other fields. Specifically, the time
evolution of a system of biochemical reactions are modelled either
deterministically, by means of a system of ordinary differential
equations, or stochastically, by means of a continuous time Markov
chain. It is natural to wonder whether the dynamics of the two modelling
regimes are linked, and whether properties of one model can shed light
on the behavior of the other one. In this talk some connections will be
Interacting particle systems is a recently developed field in
the theory of Markov processes with many applications: particle systems
have been used to model phenomena ranging from traffic behaviour to spread
of infection and tumour growth. We introduce this field through the study
of the simple exclusion process. We will construct the generator of this
process and we will give a convergence result of the spatial particle
density to the solution of the heat equation. We will also discuss a
Paths of some stochastic processes such as fractional Brownian Motion have some amazing regularization properties. It is well known that in order to have uniqueness in differential systems such as
dy_t = b(y_t) dt,
In this talk, we will present a probabilistic representation
formula for the Navier-Stokes equations on compact Riemannian manifolds.
Such a formula has been provided by Constantin and Iyer in the flat
case. On a Riemannian manifold, there are several different choices of
Laplacian operators acting on vector fields. We shall use the de
Rham-Hodge Laplacian operator which seems more relevant to the
probabilistic setting, and adopt Elworthy-Le Jan-Li's idea to decompose
it as a sum of the square of Lie derivatives. This is a joint work with
In this talk we study a non-strictly hyperbolic system of conservation law by stochastic perturbation. We show existence and uniqueness of the solution. We do not assume BV-regularity for the initial conditions. The proofs are based on the concept of entropy solution and on the method of charactteristics (under the influence of noise). This is the first result on the regularization by noise in hyperbolic systems of conservation law.
In my ongoing work with J. Quastel we consider spatially periodic growth models built from weakly asymmetric exclusion processes with finite jump ranges and general jump rates. We prove that at a large scale and after renormalization these processes converge to the Hopf-Cole solution of the KPZ equation driven by Gaussian space-time white noise. In contrast to the celebrated result by L. Bertini and G. Giacomin (in the case of the nearest neighbour interaction) and its extension by A. Dembo and L.-C.
Abstract: A ferromagnetic material possesses a magnetization, which, out of equilibrium, satisfies the Landau-Lifshitz-Gilbert equation (LLG). Thermal fluctuations are taken into account by Gaussian space-time white noise.