Regularization by noise and stochastic Burgers equations

We study a generalized 1d periodic SPDE of Burgers type: $$\partial_t u =- A^\theta u + \partial_x u^2 + A^{\theta/2} \xi$$ where $\theta > 1/2$, $-A$ is the 1d Laplacian, $\xi$ is a space-time white noise and the initial condition $u_0$ is taken to be (space) white noise. We introduce a notion of weak solution for this equation in the stationary setting. For these solutions we point out how the noise provide a regularizing effect allowing to prove existence and suitable estimates when $\theta>1/2$. When $\theta>5/4$ we obtain pathwise uniqueness. We discuss the use of the same method to study different approximations of the same equation and for a model of stationary 2d stochastic Navier-Stokes evolution.Comment: clarifications and small correction

Nonequilibrium Central Limit Theorem for a Tagged Particle in Symmetric Simple Exclusion

We prove a nonequilibirum central limit theorem for the position of a tagged particle in the one-dimensional nearest-neighbor symmetric simple exclusion process under diffusive scaling starting from a Bernoulli product measure associated to a smooth profile \rho_0:\bb R\to [0,1]

Universality of trap models in the ergodic time scale

Consider a sequence of possibly random graphs $G_N=(V_N, E_N)$, $N\ge 1$, whose vertices's have i.i.d. weights $\{W^N_x : x\in V_N\}$ with a distribution belonging to the basin of attraction of an $\alpha$-stable law, $0<\alpha<1$. Let $X^N_t$, $t \ge 0$, be a continuous time simple random walk on $G_N$ which waits a \emph{mean} $W^N_x$ exponential time at each vertex $x$. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a $K$-process. We apply this result to a class of graphs which includes the hypercube, the $d$-dimensional torus, $d\ge 2$, random $d$-regular graphs and the largest component of super-critical Erd\"os-R\'enyi random graphs