arXiv subjects

Gérard Ben-Arous

Publications and source records attributed to Gérard Ben-Arous.

3 recordsLinked to original sources

arXiv2004

Super-diffusivity in a shear flow model from perpetual homogenization

This paper is concerned with the asymptotic behavior solutions of stochastic differential equations $dy_t=dω_t -\nabla Γ(y_t) dt$, $y_0=0$ and $d=2$. $Γ$ is a $2\times 2$ skew-symmetric matrix associated to a shear flow characterized by an infinite number of spatial scales $Γ_{12}=-Γ_{21}=h(x_1)$, with $h(x_1)=\sum_{n=0}^\infty γ_n h^n(x_1/R_n)$ where $h^n$ are smooth functions of period 1, $h^n(0)=0$, $γ_n$ and $R_n$ grow exponentially fast with $n$. We can show that $y_t$ has an anomalous fast behavior ($\E[|y_t|^2]\sim t^{1+ν}$ with $ν>0$) and obtain quantitative estimates on the anomaly using and developing the tools of homogenization.

math.PR↗

arXiv2003

Aging and spin-glass dynamics

We survey the recent mathematical results about aging in certain simple disordered models. We start by the Bouchaud trap model. We then survey the results obtained for simple models of spin-glass dynamics, like the REM (the Random Energy Model, which is well approximated by the Bouchaud model on the complete graph), then the spherical Sherrington-Kirkpatrick model. We will insist on the differences in phenomenology for different types of aging in different time scales and different models. This talk is based on joint works with A.Bovier, J.Cerny, A.Dembo, V.Gayrard, A.Guionnet, as well as works by C.Newman, R.Fontes, M.Isopi, D.Stein.

math.PR↗

arXiv2002

Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion

We show that the effective diffusivity matrix $D(V^n)$ for the heat operator $\partial_t-(Δ/2-\nabla V^n \nabla)$ in a periodic potential $V^n=\sum_{k=0}^n U_k(x/R_k)$ obtained as a superposition of Holder-continuous periodic potentials $U_k$ (of period $\T^d:=\R^d/\Z^d$, $d\in \N^*$, $U_k(0)=0$) decays exponentially fast with the number of scales when the scale-ratios $R_{k+1}/R_k$ are bounded above and below. From this we deduce the anomalous slow behavior for a Brownian Motion in a potential obtained as a superposition of an infinite number of scales: $dy_t=dω_t -\nabla V^\infty(y_t) dt$

math.PR↗