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Davar Khoshnevisan

Publications and source records attributed to Davar Khoshnevisan.

At least 19 recordsLinked to original sources
arXiv2026

Points of slow growth for parabolic SPDEs

Consider the stochastic PDE, $\partial_tu = \partial^2_x u + σ(u) \dot{W}$ on $\mathbb{R}_+\times\mathbb{R}$, subject to $u(0)\equiv1$, where $\dot{W}$ denotes space-time white noise on $\mathbb{R}_+\times\mathbb{R}$ and $σ:\mathbb{R}\to\mathbb{R}$ is Lipschitz continuous. It is known that $u(t\,,x)-1$ has approximately a Gaussian distribution for every $x$ when $t\approx0$. Here we prove that there exist random points $x\in\mathbb{R}$ where the fluctuations of the solution near times zero are almost surely of sharp order $t^{1/4}$. Our work bears some loose resemblance to the study of the slow points of Brownian motion increments, though significant challenges arise due to the infinite-dimensional nature of the present problem.

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arXiv2026

On the slow points of fractional Brownian motion

Esser and Loosveldt have recently resolved a long-standing open problem in the folklore by proving that fractional Brownian motion (fBm) has slow points in the sense of Kahane, following a rich theory of slow points developed for Brownian motion and other, related, self-similar Markov processes. We presently introduce another method for the study of slow points in order to compute the Hausdorff dimension of fBm slow points. Our method follows recent ideas on the points of slow growth for SPDEs but also requires a number of new localization ideas that are likely to have other applications.

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arXiv2026

On the local well-posedness of randomly forced reaction-diffusion equations with $L^2$ initial data and a superlinear reaction term

We consider a parabolic stochastic partial differential equation (SPDE) on $[0\,,1]$ that is forced with multiplicative space-time white noise with a bounded and Lipschitz diffusion coefficient and a drift coefficient that is locally Lipschitz and satisfies an $L\log L$ growth condition. We prove that the SPDE is well posed when the initial data is in $L^2[0\,,1]$. This solves a strong form of an open problem.

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arXiv2026

The ergodic theory of SPDEs in a weak-noise regime

Consider a parabolic SPDE \[ \partial_t u = Δu + σ(u)η, \] on $(0\,,\infty)\times\mathbb{R}^d$, where $η$ is a centered, generalized Gaussian noise with $\text{Cov}[η(t\,,x)\,,η(s\,,y)]=δ_0(t-s)Λ(x-y)$ for a tempered Borel measure $Λ$ that is positive definite and satisfies a mild weak-noise. The existence of invariant measures of versions of these types of SPDEs has been studied at great length, particularly in the ``weak-noise regime''; see for example Assing and Manthey \cite{AssingManthey2003}, Chen and Eisenberg \cite{ChenEisenberg2024}, Chen, Ouyang, Tindel, and Xia \cite{ChenOuyangTindelXia2024}, Eckmann and Hairer \cite{EckmannHairer2001}, Misiats and Stanzhytskyi \cite{MSY2020}, Yu Gu and Jiawei Li \cite{GuLi2020}, and Tessitore and Zabczyk \cite{TessitoreZabczyk1998}. Here, we characterize all annealed, ergodic, invariant measures for the above SPDE in the weak-noise regime.

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arXiv2025

Uniform dimension theorems for parabolic SPDEs

Consider the following $p$-dimensional system of Itô type stochastic PDEs, \begin{align*}\left[\begin{aligned} &\partial_t u(t\,,x) = \partial^2_x u(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(t\,,x)\\ &\text{for $(t\,,x)\in(0\,,\infty)\times\mathbb{T}$, subject to $u(0) \equiv u_0$ on $\mathbb{T}$}, \end{aligned}\right.\end{align*} where $\mathbb{T}$ denotes a given one-dimensional torus, the initial data $u_0:\mathbb{T}\to\mathbb{R}^p$ is assumed to be fixed and non-random and in $C^{1/2}(\mathbb{T}\,;\mathbb{R}^p)$, and $ξ$ denotes a $p$-dimensional space-time white noise. Under certain regularity conditions on $b$ and $σ$, it is proved that, if $p \ge 4$, then \begin{equation*} \mathrm{P}\{\operatorname{dim_{_H}} u(\{t\}\times F) = 2\operatorname{dim_{_H}} F \ \text{$\forall$compact $F\subset\mathbb{T}$, $t>0$}\}=1. \end{equation*} If in addition the matrix $σ(v)$ does not depend on $v\in\mathbb{R}^p$, and is nonsingular, then the above equality holds for all $p\ge2$.

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arXiv2025

On the passage times of self-similar Gaussian processes on curved boundaries

Let $T_{c,β}$ denote the smallest $t\ge1$ that a continuous, self-similar Gaussian process with self-similarity index $α>0$ moves at least $\pm c t^β$ units. We prove that: (i) If $β>α$, then $T_{c,β}=\infty$ with positive probability; (ii) If $β<α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then $T_{c,β}$ has moments of all order; and (iii) If $β=α$ and $X$ is strongly locally nondeterministic in the sense of Pitt (1978), then there exists a continuous, strictly decreasing function $λ:(0\,,\infty)\to(0\,,\infty)$ such that $\mathrm{E}(T_{c,β}^μ)$ is finite when $0<μ<λ(c)$ and infinite when $μ>λ(c)$. Together these results extend a celebrated theorem of Breiman (1967) and Shepp (1967) for passage times of a Brownian motion on the critical square-root boundary. We briefly discuss two examples: One about fractional Brownian motion, and another about a family of linear stochastic partial differential equations.

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arXiv2025

On the well-posedness of SPDEs with locally Lipschitz coefficients

We consider the stochastic partial differential equation, $\partial_t u = \tfrac12 \partial^2_x u + b(u) + σ(u) \dot{W},$ where $u=u(t\,,x)$ is defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition $u(0)$ is bounded and measurable, and $b$ and $σ$ are locally Lipschitz continuous functions and have at most linear growth. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The results naturally generalize to the case where $b$ and $σ$ are time dependent with uniform-in-time growth and oscillation properties. Additionally, our method can be extended to the stochastic wave equation.

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arXiv2025

An Invariance Principle for some Reaction-Diffusion Equations with a Multiplicative Random Source

We establish a notion of universality for the parabolic Anderson model via an invariance principle for a wide family of parabolic stochastic partial differential equations. We then use this invariance principle in order to provide an asymptotic theory for a wide class of non-linear SPDEs. A novel ingredient of this invariance principle is the dissipativity of the underlying stochastic PDE.

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arXiv2023

Small-ball constants, and exceptional flat points of SPDEs

We study small-ball probabilities for the stochastic heat equation with multiplicative noise in the moderate-deviations regime. We prove the existence of a small-ball constant and related it to other known quantities in the literature. These small-ball estimates are known to imply Chung-type laws of the iterated logarithm (LIL) at typical spatial points; these points can be thought of as "points of flat growth". For this result in a similar context in SPDEs see, for example, the recent work of Chen \cite{Ch2023}. We establish the existence of a new family of exceptional spatial points where the Chung-type LIL fails.

math.PR↗
arXiv2023

Optimal regularity of SPDEs with additive noise

The sample-function regularity of the random-field solution to a stochastic partial differential equation (SPDE) depends naturally on the roughness of the external noise, as well as on the properties of the underlying integro-differential operator that is used to define the equation. In this paper, we consider parabolic and hyperbolic SPDEs on $0,\infty)\times\mathbb{R}^d$ of the form $\partial_t u = L u + g(u) + \dot{F} \qquad\text{and}\qquad \partial^2_t u = L u + c + \dot{F}, $ with suitable initial data, forced with a space-time homogeneous Gaussian noise $\dot{F}$ that is white in its time variable and correlated in its space variable, and driven by the generator $L$ of a genuinely $d$-dimensional Lévy process $X$. We find optimal Hölder conditions for the respective random-field solutions to these SPDEs. Our conditions are stated in terms of indices that describe thresholds on the integrability of some functionals of the characteristic exponent of the process $X$ with respect to the spectral measure of the spatial covariance of $\dot F$. Those indices are suggested by references [45, 46] on the particular case that $L$ is the Laplace operator on $\mathbb{R}^d$.

math.PR↗
arXiv2023

Instantaneous everywhere-blowup of parabolic SPDEs

We consider the following stochastic heat equation \begin{equation*} \partial_t u(t\,,x) = \tfrac12 \partial^2_x u(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) \dot{W}(t\,,x), \end{equation*} defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, where $\dot{W}$ denotes space-time white noise. The function $σ$ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function $b$ is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition \[ \int_1^\infty\frac{\mathrm{d} y}{b(y)}<\infty \] implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that $\mathbb{P}\{ u(t\,,x)=\infty\quad\text{for all $t>0$ and $x\in\mathbb{R}$}\}=1.$ The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 4.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al [3,4].

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arXiv2023

Spatial ergodicity and central limit theorems for parabolic Anderson model with delta initial condition

Let $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}}$ denote the solution to the parabolic Anderson model with initial condition $δ_0$ and driven by space-time white noise on $\mathbb{R}_+\times\mathbb{R}$, and let $p_t(x):= (2πt)^{-1/2}\exp\{-x^2/(2t)\}$ denote the standard Gaussian heat kernel on the line. We use a non-trivial adaptation of the methods in our companion papers \cite{CKNP,CKNP_b} in order to prove that the random field $x\mapsto u(t\,,x)/p_t(x)$ is ergodic for every $t >0$. And we establish an associated quantitative central limit theorem following the approach based on the Malliavin-Stein method introduced in Huang, Nualart, and Viitasaari \cite{HNV2018}.

math.PR↗
arXiv2022

On the valleys of the stochastic heat equation

We consider a generalization of the parabolic Anderson model driven by space-time white noise, also called the stochastic heat equation, on the real line. High peaks of solutions have been extensively studied under the name of intermittency, but less is known about spatial regions between peaks, which may loosely refer to as valleys. We present two results about the valleys of the solution. Our first theorem provides information about the size of valleys and the supremum of the solution over a valley. More precisely, we show that the supremum of the solution over a valley vanishes as $t\to\infty$, and we establish an upper bound of $\exp\{-\text{const}\cdot t^{1/3}\}$ for the rate of decay. We demonstrate also that the length of a valley grows at least as $\exp\{+\text{const}\cdot t^{1/3}\}$ as $t\to\infty$. Our second theorem asserts that the length of the valleys are eventually infinite when the initial data has subgaussian tails.

math.PR↗
arXiv2022

Dissipation in Parabolic SPDEs II: Oscillation and decay of the solution

We consider a stochastic heat equation of the type, $\partial_t u = \partial^2_x u + σ(u)\dot{W}$ on $(0\,,\infty)\times[-1\,,1]$ with periodic boundary conditions and on-degenerate positive initial data, where $σ:\mathbb{R} \to\mathbb{R}$ is a non-random Lipschitz continuous function and $\dot{W}$ denotes space-time white noise. If additionally $σ(0)=0$ then the solution is known to be strictly positive; see Mueller '91. In that case, we prove that the oscillation of the logarithm of the solution decays sublinearly as time tends to infinity. Among other things, it follows that, with probability one, all limit points of $t^{-1}\, \sup_{x\in[-1,1]}\, \log u(t\,,x)$ and $t^{-1}\, \inf_{x\in[-1,1]}\, \log u(t\,,x)$ must coincide. As a consequence of this fact, we prove that, when $σ$ is linear, there is a.s. only one such limit point and hence the entire path decays almost surely at an exponential rate.

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arXiv2021

Central limit theorems for parabolic stochastic partial differential equations

Let $\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel measure $f$ and satisfies Dalang's condition. We prove two general functional central limit theorems for occupation fields of the form $N^{-d} \int_{\mathbb{R}^d} g(u(t\,,x)) ψ(x/N)\, \mathrm{d} x$ as $N\rightarrow \infty$, where $g$ runs over the class of Lipschitz functions on $\mathbb{R}^d$ and $ψ\in L^2(\mathbb{R}^d)$. The proof uses Poincaré-type inequalities, Malliavin calculus, compactness arguments, and Paul Lévy's classical characterization of Brownian motion as the only mean zero, continuous Lévy process. Our result generalizes central limit theorems of Huang et al \cite{HuangNualartViitasaari2018,HuangNualartViitasaariZheng2019} valid when $g(u)=u$ and $ψ= \mathbf{1}_{[0,1]^d}$.

math.PR↗
arXiv2021

Spatial stationarity, ergodicity and CLT for parabolic Anderson model with delta initial condition in dimension $d\geq 1$

Suppose that $\{u(t\,, x)\}_{t >0, x \in\mathbb{R}^d}$ is the solution to a $d$-dimensional parabolic Anderson model with delta initial condition and driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance given by a nonnegative-definite measure $f$ which satisfies Dalang's condition. Let $\boldsymbol{p}_t(x):=(2πt)^{-d/2}\exp\{-\|x\|^2/(2t)\}$ denote the standard Gaussian heat kernel on $\mathbb{R}^d$. We prove that for all $t>0$, the process $U(t):=\{u(t\,, x)/\boldsymbol{p}_t(x): x\in \mathbb{R}^d\}$ is stationary using Feynman-Kac's formula, and is ergodic under the additional condition $\hat{f}\{0\}=0$, where $\hat{f}$ is the Fourier transform of $f$. Moreover, using Malliavin-Stein method, we investigate various central limit theorems for $U(t)$ based on the quantitative analysis of $f$. In particular, when $f$ is given by Riesz kernel, i.e., $f(\mathrm{d} x) = \|x\|^{-β}\mathrm{d} x$, we obtain a multiple phase transition for the CLT for $U(t)$ from $β\in(0\,,1)$ to $β=1$ to $β\in(1\,,d\wedge 2)$.

math.PR↗
arXiv2020

Phase Analysis for a family of Stochastic Reaction-Diffusion Equations

We consider a reaction-diffusion equation of the type \[ \partial_tψ= \partial^2_xψ+ V(ψ) + λσ(ψ)\dot{W} \qquad\text{on $(0\,,\infty)\times\mathbb{T}$}, \] subject to a "nice" initial value and periodic boundary, where $\mathbb{T}=[-1\,,1]$ and $\dot{W}$ denotes space-time white noise. The reaction term $V:\mathbb{R}\to\mathbb{R}$ belongs to a large family of functions that includes Fisher--KPP nonlinearities [$V(x)=x(1-x)$] as well as Allen-Cahn potentials [$V(x)=x(1-x)(1+x)$], the multiplicative nonlinearity $σ:\mathbb{R}\to\mathbb{R}$ is non random and Lipschitz continuous, and $λ>0$ is a non-random number that measures the strength of the effect of the noise $\dot{W}$. The principal finding of this paper is that: (i) When $λ$ is sufficiently large, the above equation has a unique invariant measure; and (ii) When $λ$ is sufficiently small, the collection of all invariant measures is a non-trivial line segment, in particular infinite. This proves an earlier prediction of Zimmerman et al. (2000). Our methods also say a great deal about the structure of these invariant measures.

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