arXiv subjects

Houman Owhadi

Publications and source records attributed to Houman Owhadi.

113 records · Page 7Linked to original sources

arXiv2004

Anomalous Slow Diffusion from Perpetual Homogenization

This paper is concerned with the asymptotic behavior of solutions of stochastic differential equations $dy_t=dω_t -\nabla V(y_t) dt$, $y_0=0$. When $d=1$ and $V$ is not periodic but obtained as a superposition of an infinite number of periodic potentials with geometrically increasing periods ($V(x) = \sum_{k=0}^\infty U_k(x/R_k)$, where $U_k$ are smooth functions of period 1, $U_k(0)=0$, and $R_k$ grows exponentially fast with $k$) we can show that $y_t$ has an anomalous slow behavior and we obtain quantitative estimates on the anomaly using and developing the tools of homogenization. Pointwise estimates are based on a new analytical inequality for sub-harmonic functions. When $d\geq 1$ and $V$ is periodic, quantitative estimates are obtained on the heat kernel of $y_t$, showing the rate at which homogenization takes place. The latter result proves Davies's conjecture and is based on a quantitative estimate for the Laplace transform of martingales that can be used to obtain similar results for periodic elliptic generators

math.PR↗

arXiv2004

Approximation of the effective conductivity of ergodic media by periodization

This paper is concerned with the approximation of the effective conductivity $σ(A,μ)$ associated to an elliptic operator $\nabla_x A(x,η) \nabla_x$ where for $x\in \R^d$, $d\geq 1$, $A(x,η)$ is a bounded elliptic random symmetric $d\times d$ matrix and $η$ takes value in an ergodic probability space $(X,μ)$. Writing $A^N(x,η)$ the periodization of $A(x,η)$ on the torus $T^d_N$ of dimension $d$ and side $N$ we prove that for $μ$-almost all $η$ $$ \lim_{N\to +\infty}σ(A^N,η)=σ(A,μ) $$ We extend this result to non-symmetric operators $\nabla_x (a+E(x,η)) \nabla_x$ corresponding to diffusions in ergodic divergence free flows ($a$ is $d\times d$ elliptic symmetric matrix and $E(x,η)$ an ergodic skew-symmetric matrix); and to discrete operators corresponding to random walks on $\Z^d$ with ergodic jump rates. The core of our result is to show that the ergodic Weyl decomposition associated to $Ł^2(X,μ)$ can almost surely be approximated by periodic Weyl decompositions with increasing periods, implying that semi-continuous variational formulae associated to $Ł^2(X,μ)$ can almost surely be approximated by variational formulae minimizing on periodic potential and solenoidal functions.

math.PR↗

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↗

arXiv2004

Averaging versus Chaos in Turbulent Transport?

In this paper we analyze the transport of passive tracers by deterministic stationary incompressible flows which can be decomposed over an infinite number of spatial scales without separation between them. It appears that a low order dynamical system related to local Peclet numbers can be extracted from these flows and it controls their transport properties. Its analysis shows that these flows are strongly self-averaging and super-diffusive: the delay $τ(r)$ for any finite number of passive tracers initially close to separate till a distance $r$ is almost surely anomalously fast ($τ(r)\sim r^{2-ν}$, with $ν>0$). This strong self-averaging property is such that the dissipative power of the flow compensates its convective power at every scale. However as the circulation increase in the eddies the transport behavior of the flow may (discontinuously) bifurcate and become ruled by deterministic chaos: the self-averaging property collapses and advection dominates dissipation. When the flow is anisotropic a new formula describing turbulent conductivity is identified.

math-ph↗

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↗