Large deviations for Brownian motion in a random scenery

We prove large deviations principles in large time, for the Brownian occupation time in random scenery. The random scenery is constant on unit cubes, and consist of i.i.d. bounded variables, independent of the Brownian motion. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched'' and ``annealed'' settings.Comment: 29 page

Quenched large deviations for diffusions in a random Gaussian shear flow drift

We prove a full large deviations principle in large time, for a diffusion process with random drift V, which is a centered Gaussian shear flow random field. The large deviations principle is established in a ``quenched'' setting, i.e. is valid almost surely in the randomness of V.Comment: 29 page

Regularity of quasi-stationary measures for simple exclusion in dimension d >= 5

We consider the symmetric simple exclusion process on Z^d, for d>= 5, and study the regularity of the quasi-stationary measures of the dynamics conditionned on not occupying the origin. For each \rho\in ]0,1[, we establish uniqueness of the density of quasi-stationary measures in L^2(d\nur), where \nur is the stationary measure of density \rho. This, in turn, permits us to obtain sharp estimates for P_{\nur}(\tau>t), where \tau is the first time the origin is occupied.Comment: 18 pages. Corrections after referee report. To be published in Ann Proba

Hitting times for independent random walks on Zd\mathbb{Z}^d

We consider a system of asymmetric independent random walks on $\mathbb{Z}^d$, denoted by $\{\eta_t,t\in{\mathbb{R}}\}$, stationary under the product Poisson measure $\nu_{\rho}$ of marginal density $\rho>0$. We fix a pattern $\mathcal{A}$, an increasing local event, and denote by $\tau$ the hitting time of $\mathcal{A}$. By using a loss network representation of our system, at small density, we obtain a coupling between the laws of $\eta_t$ conditioned on $\{\tau>t\}$ for all times $t$. When $d\ge3$, this provides bounds on the rate of convergence of the law of $\eta_t$ conditioned on $\{\tau>t\}$ toward its limiting probability measure as $t$ tends to infinity. We also treat the case where the initial measure is close to $\nu_{\rho}$ without being product.Comment: Published at http://dx.doi.org/10.1214/009117906000000106 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Self-Diffusion in Simple Models: Systems with Long-Range Jumps

We review some exact results for the motion of a tagged particle in simple models. Then, we study the density dependence of the self diffusion coefficient, $D_N(\rho)$, in lattice systems with simple symmetric exclusion in which the particles can jump, with equal rates, to a set of $N$ neighboring sites. We obtain positive upper and lower bounds on $F_N(\rho)=N((1-\r)-[D_N(\rho)/D_N(0)])/(\rho(1-\rho))$ for $\rho\in [0,1]$. Computer simulations for the square, triangular and one dimensional lattice suggest that $F_N$ becomes effectively independent of $N$ for $N\ge 20$.Comment: 24 pages, in TeX, 1 figure, e-mail addresses: [email protected], [email protected], [email protected]

Annealed lower tails for the energy of a polymer

We consider the energy of a randomly charged polymer. We assume that only charges on the same site interact pairwise. We study the lower tails of the energy, when averaged over both randomness, in dimension three or more. As a corollary, we obtain the correct temperature-scale for the Gibbs measure.Comment: 27 page

Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.Comment: 25 page

Diffusion Effects on the Breakdown of a Linear Amplifier Model Driven by the Square of a Gaussian Field

We investigate solutions to the equation $\partial_t{\cal E} - {\cal D}\Delta {\cal E} = \lambda S^2{\cal E}$, where $S(x,t)$ is a Gaussian stochastic field with covariance $C(x-x',t,t')$, and $x\in {\mathbb R}^d$. It is shown that the coupling $\lambda_{cN}(t)$ at which the $N$-th moment diverges at time $t$, is always less or equal for ${\cal D}>0$ than for ${\cal D}=0$. Equality holds under some reasonable assumptions on $C$ and, in this case, $\lambda_{cN}(t)=N\lambda_c(t)$ where $\lambda_c(t)$ is the value of $\lambda$ at which diverges. The ${\cal D}=0$ case is solved for a class of $S$. The dependence of $\lambda_{cN}(t)$ on $d$ is analyzed. Similar behavior is conjectured when diffusion is replaced by diffraction, ${\cal D}\to i{\cal D}$, the case of interest for backscattering instabilities in laser-plasma interaction.Comment: 19 pages, in LaTeX, e-mail addresses: [email protected], [email protected], [email protected], [email protected]

Self-intersection local times of random walks: Exponential moments in subcritical dimensions

Fix $p>1$, not necessarily integer, with $p(d-2)<d$. We study the $p$-fold self-intersection local time of a simple random walk on the lattice $\Z^d$ up to time $t$. This is the $p$-norm of the vector of the walker's local times, $\ell_t$. We derive precise logarithmic asymptotics of the expectation of $\exp\{\theta_t \|\ell_t\|_p\}$ for scales $\theta_t>0$ that are bounded from above, possibly tending to zero. The speed is identified in terms of mixed powers of $t$ and $\theta_t$, and the precise rate is characterized in terms of a variational formula, which is in close connection to the {\it Gagliardo-Nirenberg inequality}. As a corollary, we obtain a large-deviation principle for $\|\ell_t\|_p/(t r_t)$ for deviation functions $r_t$ satisfying t r_t\gg\E[\|\ell_t\|_p]. Informally, it turns out that the random walk homogeneously squeezes in a $t$-dependent box with diameter of order $\ll t^{1/d}$ to produce the required amount of self-intersections. Our main tool is an upper bound for the joint density of the local times of the walk.Comment: 15 pages. To appear in Probability Theory and Related Fields. The final publication is available at springerlink.co