Brownian yet non-Gaussian diffusion through equilibrium nonlinear friction

Large deviations, gradient flows and fluctuating hydrodynamics for fractional diffusion

This work is a step towards a general understanding of fluctuations for systems of particles described at large scale by fractional diffusion. First, we derive formally a Large Deviation Principle for the empirical measure of mean-field interacting particles with a L{\'e}vy-type noise. In the case of independent partic...

Anomalous scaling limit of a Brownian particle in a log-correlated potential

We study a Brownian particle in $\mathbb{R}^d$, $d\geq2$, with drift given by the gradient of a log-correlated Gaussian potential. At weak disorder, we prove convergence to a scaling limit whose law is singular with respect to Brownian motion and whose invariant measure is given by Gaussian multiplicative chaos. The pr...

Stochastic fractional diffusion equations with spatial Gaussian noise in the Stratonovich regime

For a class of stochastic fractional diffusion equations, we prove the existence of Stratonovich solutions and obtain the precise values of the Lyapunov exponents governing the large $t$-asymptotics of their $p$-th moments for all real $p\in\{1\}\cup[2,\infty)$, as well as the large $p$-asymptotics of the $p$-th moment...

Non-negative differential mobility of one-dimensional underdamped Langevin dynamics: Proof via mapping to a marginally stable oscillator

For an underdamped particle driven by a constant non-conservative force across a periodic potential in one spatial dimension at finite temperature, numerical studies have shown that the differential mobility is non-negative. Here, an analytical proof will be given that is based on expressing the differential mobility b...

Asymptotic Behavior of the Smoluchowski–Kramers Approximation for Distribution-Dependent Stochastic Differential Equations Driven by Fractional Brownian Motion

Fractional Brownian motion with multivariate time: persistence exponents

We consider fractional Brownian motion with D-dimensional time in the domain G(T)=(-T<ti<T,tj>0 if j<d+1). Let P(T) be the probability that the process will not exceed a fixed level in G(T). We show that for large T, logP(T)=(D-dH)logT(1+o(1)) where H is the Hurst parameter of the process