We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j \geq 0$, with $\sum_{j=1}^n x_j = E$. For some $ε$, the allowed configurations $(x_1,\dots,x_n)$ are those that satisfy $|x_i - x_j| \geq ε$ for all $i\neq j$. At each step of the process, a pair $(i,j)$ of particles is selected uniformly at random, and then they "collide", and there is a repartition of their total energy $x_i + x_j$ between them producing new energies $x^*_i$ and $x^*_j$ with $x^*_i + x^*_j = x_i + x_j$, but with the restriction that exclusion rule is still observed for the new pair of energies. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the "non-quantized" exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a detailed notion of Kac's chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.
In this article we obtain the equilibrium fluctuations of a symmetric exclusion process in $\mathbb{Z}$ with long jumps. The transition probability of the jump from $x$ to $y$ is proportional to $|x-y|^{-γ-1}$. Here we restrict to the choice $γ\geq 2$ so that the system has a diffusive behavior. Moreover, when particles move between $\mathbb{Z}_{-}^{*}$ and $\mathbb N$, the jump rates are slowed down by a factor $αn^{-β}$, where $α>0$, $β\geq 0$ and $n$ is the scaling parameter. Depending on the values of $β$ and $γ$, we obtain several stochastic partial differential equations, corresponding t…
We analyze the equilibrium fluctuations of the density, current and tagged particle in symmetric exclusion with a slow bond. The system evolves in the one-dimensional lattice and the jump rate is everywhere equal to one except at the slow bond where it is $αn^-β$, where $α,β\geq{0}$ and $n$ is the scaling parameter. Depending on the regime of $β$, we find three different behaviors for the limiting fluctuations whose covariances are explicitly computed. In particular, for the critical value $β=1$, starting a tagged particle near the slow bond, we obtain a family of gaussian processes indexed in…
We consider exclusion processes on a rooted $d$-regular tree. We start from a Bernoulli product measure conditioned on having a particle at the root, which we call the tagged particle. For $d\geq 3$, we show that the tagged particle has positive linear speed and satisfies a central limit theorem. We give an explicit formula for the speed. As a key step in the proof, we first show that the exclusion process "seen from the tagged particle" has an ergodic invariant measure.
We investigate signatures of convergence for a sequence of diffusion processes on a line, in conservative force fields stemming from superharmonic potentials $U(x)\sim x^m$, $m=2n \geq 2$. This is paralleled by a transformation of each $m$-th diffusion generator $L = DΔ+ b(x)\nabla $, and likewise the related Fokker-Planck operator $L^*= DΔ- \nabla [b(x)\, \cdot]$, into the affiliated Schrödinger one $\hat{H}= - DΔ+ {\cal{V}}(x)$. Upon a proper adjustment of operator domains, the dynamics is set by semigroups $\exp(tL)$, $\exp(tL_*)$ and $\exp(-t\hat{H})$, with $t \geq 0$. The Feynman-Kac inte…
We consider the (discrete) parabolic Anderson model $\partial u(t,x)/\partial t=Δu(t,x) +ξ_t(x) u(t,x)$, $t\geq 0$, $x\in \mathbb{Z}^d$, where the $ξ$-field is $\mathbb{R}$-valued and plays the role of a dynamic random environment, and $Δ$ is the discrete Laplacian. We focus on the case in which $ξ$ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein--Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension $d=3$ upon considering a suitably renormalised version of the above equati…
We study a new model of interactive particle systems which we call the randomly activated cascading exclusion process (RACEP). Particles wake up according to exponential clocks and then take a geometric number of steps. If another particle is encountered during these steps, the first particle goes to sleep at that location and the second is activated and proceeds accordingly. We consider a totally asymmetric version of this model which we refer as Hall-Littlewood-PushTASEP (HL-PushTASEP) on $\mathbb{Z}_{\geq 0}$ lattice where particles only move right and where initially particles are distribu…
We consider the weakly asymmetric simple exclusion process in the presence of a slow bond and starting from the invariant state, namely the Bernoulli product measure of parameter $ρ\in(0,1)$. The rate of passage of particles to the right (resp. left) is $\frac1{\vphantom{n^β}2}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\frac1{\vphantom{n^β}2}-\frac{a}{2n^{\vphantomβγ}}$) except at the bond of vertices $\{-1,0\}$ where the rate to the right (resp. left) is given by $\fracα{2n^β}+\frac{a}{2n^{\vphantomβγ}}$ (resp. $\fracα{2n^β}-\frac{a}{2n^{\vphantomβγ}}$). Above, $α>0$, $γ\geq β\geq 0$, $a\geq 0$. Fo…
We consider a random walk in dimension $d\geq 1$ in a dynamic random environment evolving as an interchange process with rate $γ>0$. We only assume that the annealed drift is non-zero. We prove that the empirical velocity of the walker $X_t/t$ eventually lies in an arbitrary small ball around the annealed drift if we choose $γ$ large enough. This statement is thus a perturbation of the case $γ=+\infty$ where the environment is refreshed between each step of the walker. We extend three-way part of the results of HS15, where the environment was given by the $1$ dimensional exclusion process: $(i…
This work concerns the Ornstein-Uhlenbeck type process associated to a positive self-similar Markov process $(X(t))_{t\geq 0}$ which drifts to $\infty$, namely $U(t):= {\rm e}^{-t}X({\rm e}^t-1)$. We point out that $U$ is always a (topologically) recurrent Markov process and identify its invariant measure in terms of the law of the exponential functional $\hat I := \int_0^\infty \exp(\hatξ_s) {\rm d}s$, where $\hatξ$ is the dual of the real-valued Lévy process $ξ$ related to $X$ by the Lamperti transformation. This invariant measure is infinite (i.e. $U$ is null-recurrent) if and only if $ξ_1\…
