arXiv subjects

Davar Khoshnevisan

Publications and source records attributed to Davar Khoshnevisan.

At least 37 records · Page 2Linked to original sources

arXiv2020

Central limit theorems for spatial averages of the stochastic heat equation via Malliavin-Stein's method

Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang's condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form $N^{-d} \int_{[0,N]^d} g(u(t\,,x))\, \mathrm{d} x$, as $N\rightarrow\infty$, where $g$ is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein's method for normal approximations. Our results include a central limit theorem for the {\it Hopf-Cole} solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

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arXiv2019

Spatial ergodicity for SPDEs via Poincaré-type inequalities

Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs \cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of such discussions. Our methods hinge on novel facts from harmonic analysis and functions of positive type, as well as from Malliavin calculus and Poincaré inequalities. We further showcase the utility of these Poincaré inequalities by: (a) describing conditions that ensure that the random field $u(t)$ is mixing for every $t>0$; and by (b) giving a quick proof of a conjecture of Conus et al \cite{CJK12} about the "size" of the intermittency islands of $u$. The ergodicity and the mixing results of this paper are sharp, as they include the classical theory of Maruyama \cite{Maruyama} (see also Dym and McKean \cite{DymMcKean}) in the simple setting where the nonlinear term $σ$ is a constant function.

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arXiv2019

Spatial ergodicity for SPDEs via a Poincaré-type inequality

Consider a parabolic stochastic PDE of the form $\partial_t u=\frac{1}{2}Δu + σ(u)η$, where $u=u(t\,,x)$ for $t\ge0$ and $x\in\mathbb{R}^d$, $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous and non random, and $η$ is a centered Gaussian noise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)\equiv1$, then we prove that, under a mild decay condition on $f$, the process $x\mapsto u(t\,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs \cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.

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arXiv2019

Dissipation in parabolic SPDEs

The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.

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arXiv2018

Talagrand Concentration Inequalities for Stochastic Partial Differential Equations

One way to define the concentration of measure phenomenon is via Talagrand inequalities, also called transportation-information inequalities. That is, a comparison of the Wasserstein distance from the given measure to any other absolutely continuous measure with finite relative entropy. Such transportation-information inequalities were recently established for some stochastic differential equations. Here, we develop a similar theory for some stochastic partial differential equations.

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arXiv2017

Analysis of a Stratified Kraichnan Flow

We consider the stochastic convection-diffusion equation \[ \partial_t u(t\,,{\bf x}) =νΔu(t\,,{\bf x}) + V(t\,,x_1)\partial_{x_2}u(t\,,{\bf x}), \] for $t>0$ and ${\bf x}=(x_1\,,x_2)\in\mathbb{R}^2$, subject to $θ_0$ being a nice initial profile. Here, the velocity field $V$ is assumed to be centered Gaussian with covariance structure \[ \text{Cov}[V(t\,,a)\,,V(s\,,b)]= δ_0(t-s)ρ(a-b)\qquad\text{for all $s,t\ge0$ and $a,b\in\mathbb{R}$}, \] where $ρ$ is a continuous and bounded positive-definite function on $\mathbb{R}$. We prove a quite general existence/uniqueness/regularity theorem, together with a probabilistic representation of the solution that represents $u$ as an expectation functional of an exogenous infinite-dimensional Brownian motion. We use that probabilistic representation in order to study the Itô/Walsh solution, when it exists, and relate it to the Stratonovich solution which is shown to exist for all $ν>0$. Our a priori estimates imply the physically-natural fact that, quite generally, the solution dissipates. In fact, very often, \begin{equation} P\left\{\sup_{|x_1|\leq m}\sup_{x_2\in\mathbb{R}} |u(t\,,{\bf x})| = O\left(\frac{1}{\sqrt t}\right)\qquad\text{as $t\to\infty$} \right\}=1\qquad\text{for all $m>0$}, \end{equation} and the $O(1/\sqrt t)$ rate is shown to be unimproveable. Our probabilistic representation is malleable enough to allow us to analyze the solution in two physically-relevant regimes: As $t\to\infty$ and as $ν\to 0$. Among other things, our analysis leads to a "macroscopic multifractal analysis" of the rate of decay in the above equation in terms of the reciprocal of the Prandtl (or Schmidt) number, valid in a number of simple though still physically-relevant cases.

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arXiv2017

On the multifractal local behavior of parabolic stochastic PDEs

Consider the stochastic heat equation $\dot{u}=\frac12 u"+σ(u)ξ$ on $(0\,,\infty)\times\mathbb{R}$ subject to $u(0)\equiv1$, where $σ:\mathbb{R}\to\mathbb{R}$ is a Lipschitz (local) function that does not vanish at $1$, and $ξ$ denotes space-time white noise. It is well known that $u$ has continuous sample functions; as a result, $\lim_{t\downarrow0}u(t\,,x)= 1$ almost surely for every $x\in\mathbb{R}$. The corresponding fluctuations are also known: For every fixed $x\in\mathbb{R}$, $t\mapsto u(t\,,x)$ looks locally like a fixed multiple of fractional Brownian motion (fBm) with index $1/4$. In particular, an application of Fubini's theorem implies that, on an $x$-set of full Lebesgue measure, the short-time behavior of the peaks of the random function $t\mapsto u(t\,,x)$ are governed by the law of the iterated logarithm for fBm, up to possibly a suitable normalization constant. By contrast, the main result of this paper claims that, on an $x$-set of full Hausdorff dimension, the short-time peaks of $t\mapsto u(t\,,x)$ follow a non-iterated logarithm law, and that those peaks contain a rich multifractal structure a.s. Large-time variations of these results were predicted in the physics literature a number of years ago and proved very recently in Khoshnevisan, Kim and Xiao (2016). To the best of our knowledge, the short-time results of the present paper are observed here for the first time.

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arXiv2017

Global solutions to reaction-diffusion equations with super-linear drift and multiplicative noise

Let $ξ(t\,,x)$ denote space-time white noise and consider a reaction-diffusion equation of the form \[ \dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(t\,,x), \] on $\mathbb{R}_+\times[0\,,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon>0$ such that $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$ for all sufficiently-large values of $|z|$. When $σ\equiv 0$, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have recently shown that there is finite-time blowup when $σ$ is a non-zero constant. In this paper, we prove that the Bonder--Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is "typically" well posed when $\vert b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.

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arXiv2017

A macroscopic multifractal analysis of parabolic stochastic PDEs

It is generally argued that the solution to a stochastic PDE with multiplicative noise---such as $\dot{u}=\frac12 u"+uξ$, where $ξ$ denotes space-time white noise---routinely produces exceptionally-large peaks that are "macroscopically multifractal." See, for example, Gibbon and Doering (2005), Gibbon and Titi (2005), and Zimmermann et al (2000). A few years ago, we proved that the spatial peaks of the solution to the mentioned stochastic PDE indeed form a random multifractal in the macroscopic sense of Barlow and Taylor (1989; 1992). The main result of the present paper is a proof of a rigorous formulation of the assertion that the spatio-temporal peaks of the solution form infinitely-many different multifractals on infinitely-many different scales, which we sometimes refer to as "stretch factors." A simpler, though still complex, such structure is shown to also exist for the constant-coefficient version of the said stochastic PDE.

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arXiv2016

Delocalization of a $(1+1)$-dimensional stochastic wave equation

A noteworthy property of many parabolic stochastic PDEs is that they locally linearize (Foondun, Khoshnevisan and Mahboubi (2015), Hairer (2013, 2014), Hairer and Pardoux (2015), Khoshnevisan, Swanson, Xiao and Zhang (2013)). We prove that, by contrast, a large family of stochastic wave equations in dimension one do not possess this important property.

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arXiv2016

On the Macroscopic Fractal Geometry of Some Random Sets

This paper is concerned mainly with the macroscopic fractal behavior of various random sets that arise in modern and classical probability theory. Among other things, it is shown here that the macroscopic behavior of Boolean coverage processes is analogous to the microscopic structure of the Mandelbrot fractal percolation. Other, more technically challenging, results of this paper include: (i) The computation of the macroscopic dimension of the graph of a large family of Lévy processes; and (ii) The determination of the macroscopic monofractality of the extreme values of symmetric stable processes. As a consequence of (i), it will be shown that the macroscopic fractal dimension of the graph of Brownian motion differs from its microscopic fractal dimension. Thus, there can be no scaling argument that allows one to deduce the macroscopic geometry from the microscopic. Item (ii) extends the recent work of Khoshnevisan, Kim, and Xiao \cite{KKX} on the extreme values of Brownian motion, using a different method.

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arXiv2015

A Boundedness Trichotomy for the Stochastic Heat Equation

We consider the stochastic heat equation with a multiplicative white noise forcing term under standard "intermitency conditions." The main finding of this paper is that, under mild regularity hypotheses, the a.s.-boundedness of the solution $x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $Λ:= \lim_{|x|\to\infty} |\log u_0(x)|/(\log|x|)^{2/3}$.

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arXiv2015

Semi-discrete semi-linear parabolic SPDEs

Consider an infinite system \[\partial_tu_t(x)=(\mathscr{L}u_t)(x)+ σ\bigl(u_t(x)\bigr)\partial_tB_t(x)\] of interacting Itô diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solution under standard regularity assumptions on the nonlinearity $σ$. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the $k$th moment Lyapunov exponent is frequently of sharp order $k^2$, in contrast to the continuous-space stochastic heat equation whose $k$th moment Lyapunov exponent can be of sharp order $k^3$. When the underlying walk is transient and the noise level is sufficiently low, we prove also that the solution is a.s. uniformly dissipative provided that the initial profile is in $\ell^1(\mathbf {Z}^d)$.

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arXiv2015

Nonlinear noise excitation of intermittent stochastic PDEs and the topology of LCA groups

Consider the stochastic heat equation $\partial_tu=\mathscr{L}u+λσ(u)ξ$, where $\mathscr{L}$ denotes the generator of a Lévy process on a locally compact Hausdorff Abelian group $G$, $σ:\mathbf{R}\to\mathbf{R}$ is Lipschitz continuous, $λ\gg1$ is a large parameter, and $ξ$ denotes space-time white noise on $\mathbf{R}_+\times G$. The main result of this paper contains a near-dichotomy for the (expected squared) energy $\mathrm{E}(\|u_t\|_{L^2(G)}^2)$ of the solution. Roughly speaking, that dichotomy says that, in all known cases where $u$ is intermittent, the energy of the solution behaves generically as $\exp\{\operatorname {const}\cdot\,λ^2\}$ when $G$ is discrete and $\ge\exp\{\operatorname {const}\cdot\,λ^4\}$ when $G$ is connected.

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arXiv2015

Intermittency and multifractality: A case study via parabolic stochastic PDEs

Let $ξ$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + uξ$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + ξ$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic PDEs on $\mathbf{R}_+\times\mathbf{R}^d$ with $d\ge 2$. G. Lawler has asked us if intermittency is the same as multifractality. The present work gives a negative answer to this question. As a byproduct of our methods, we prove also that the peaks of the Brownian motion form a large-scale monofractal, whereas the peaks of the Ornstein--Uhlenbeck process on $\mathbf{R}$ are multifractal. Throughout, we make extensive use of the macroscopic fractal theory of M.T. Barlow and S. J. Taylor (1989, 1992). We expand on aspects of the Barlow-Taylor theory, as well.

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arXiv2015

Brownian motion and thermal capacity

Let $W$ denote $d$-dimensional Brownian motion. We find an explicit formula for the essential supremum of Hausdorff dimension of $W(E)\cap F$, where $E\subset(0,\infty)$ and $F\subset \mathbf {R}^d$ are arbitrary nonrandom compact sets. Our formula is related intimately to the thermal capacity of Watson [Proc. Lond. Math. Soc. (3) 37 (1978) 342-362]. We prove also that when $d\ge2$, our formula can be described in terms of the Hausdorff dimension of $E\times F$, where $E\times F$ is viewed as a subspace of space time.

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arXiv2014

Dissipation and high disorder

Given a field $\{B(x)\}_{x\in\mathbf{Z}^d}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^d$, the generator of a suitable Markov process on $\mathbf{Z}^d,\,\,\mathcal{G},$ and sufficiently nice function $σ:[0,\infty)\to[0,\infty),$ we consider the influence of the parameter $λ$ on the behavior of the system, \begin{align*} \rm{d} u_t(x) = & (\mathcal{G}u_t)(x)\,\rm{d} t + λσ(u_t(x))\rm{d} B_t(x) \qquad[t>0,\ x\in\mathbf{Z}^d], &u_0(x)=c_0δ_0(x). \end{align*} We show that for any $λ>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for dimensions greater than two there is a phase transition point $λ_c\in(0,\infty)$ such that for $λ>λ_c,\, \sum_{\mathbf{Z}^d}u_t(x)\to 0$ as $t\to\infty$ while for $λ<λ_c,\,\sum_{\mathbf{Z}^d}u_t(x)\not\to 0$ as $t\to\infty.$

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arXiv2014

The dimension of the range of a transient random walk

We find formulas for the macroscopic Minkowski and Hausdorff dimensions of the range of an arbitrary transient walk in Z^d. This endeavor solves a problem of Barlow and Taylor (1991).

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