About

Davar Khoshnevisan is a professor in the Department of Mathematics at the University of Utah. His research focuses on stochastic partial differential equations (SPDEs), ergodic theory, and probability theory. He has contributed to the understanding of the behavior of SPDEs in various regimes, including weak-noise conditions, and has worked on topics such as the passage times of Gaussian processes, invariance principles for reaction-diffusion equations with random sources, and the well-posedness of SPDEs with locally Lipschitz coefficients. His work is supported by the National Science Foundation, and he maintains a personal web page with his publications, notes, and other writings.

Research topics

• Mathematical analysis
• Mathematics
• Humanities
• Mathematical physics
• Pure mathematics
• Physics
• Quantum mechanics
• Philosophy
• Statistics

Selected publications

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

  Probability Theory and Related Fields · 2026-04-13

  articleOpen access

  Abstract 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 $$L\log L$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>log</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> </mml:math> growth condition. We prove that the SPDE is well posed when the initial data is in $$L^2[0,1]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This solves a strong form of an open problem.

  PublisherOA PDFDOI

• On the slow points of fractional Brownian motion

  ArXiv.org · 2026-03-08

  articleOpen access1st authorCorresponding

  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.

  PublisherOA PDF

• The ergodic theory of SPDEs in a weak-noise regime

  ArXiv.org · 2026-03-19

  articleOpen access

  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.

  PublisherOA PDF

• The ergodic theory of SPDEs in a weak-noise regime

  arXiv (Cornell University) · 2026-03-19

  preprintOpen access

  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.

  PublisherDOI

• On the slow points of fractional Brownian motion

  Open MIND · 2026-03-08

  preprint1st authorCorresponding

  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.

  DOI

• On the Well-Posedness of Stochastic Partial Differential Equations with Locally Lipschitz Coefficients

  Journal of Theoretical Probability · 2026-02-23 · 1 citations

  articleOpen access

  Abstract We consider the stochastic partial differential equation (SPDE) $$\begin{aligned} \partial _t u = \tfrac{1}{2} \partial ^2_x u + b(u) + \sigma (u) \dot{W}, \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where $$u=u(t,x)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is defined for $$(t,x)\in (0,\infty )\times \mathbb {R}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> and $$\dot{W}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> 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 $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.

  PublisherOA PDFDOI

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

  arXiv (Cornell University) · 2025-09-30

  preprintOpen access

  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.

  PublisherOA PDFDOI

• An interview with Mu-Fa Chen

  Discrete and Continuous Dynamical Systems - S · 2025-01-01

  articleOpen access1st authorCorresponding

  Professor Mu-Fa Chen was an invited speaker at the 2016 Frontier Probability Days held on the campus of the University of Utah, May 9–11, 2016. Professor Chen is among China's most distinguished and influential living probabilists. He is an elected member of the Chinese Academy of Science, and fellow of the World Academy of Sciences (TWAS) and the American Mathematical Society. Together with his student F Y Wang, his research involves the development of powerful probabilistic techniques for the sharp estimation of eigenvalues of the Laplacian on manifolds having a positive lower bound on their Ricci curvature, in terms of the bound, the dimension and the diameter of the manifold. His 2005 Springer monograph Eigenvalues, Inequalities, and Ergodic Theory provides an influential testimony to his remarkable contributions to this deep and challenging area of mathematics. Having spent his entire career in China, Professor Chen's relatively infrequent excursion outside China provided a welcome opportunity to learn more about his personal and professional journey in probability over these past fifty years. The following is an interview conducted by two of the conference organizers, Davar Koshnivisan, University of Utah, and Edward Waymire, Oregon State University.

  PublisherOA PDFDOI

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

  ArXiv.org · 2025-04-15

  preprintOpen access1st authorCorresponding

  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.

  PublisherOA PDFDOI

• Instantaneous everywhere-blowup of parabolic SPDEs

  Probability Theory and Related Fields · 2024-03-05 · 7 citations

  articleOpen access

  Abstract We consider the following stochastic heat equation $$\begin{aligned} \partial _t u(t,x) = \tfrac{1}{2} \partial ^2_x u(t,x) + b(u(t,x)) + \sigma (u(t,x)) {\dot{W}}(t,x), \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> <mml:msubsup> <mml:mi>∂</mml:mi> <mml:mi>x</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> defined for $$(t,x)\in (0,\infty )\times {\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>∞</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> , where $${\dot{W}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>W</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> </mml:math> denotes space-time white noise. The function $$\sigma $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>σ</mml:mi> </mml:math> 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 $$\begin{aligned} \int _1^\infty \frac{\textrm{d}y}{b(y)}&lt;\infty \end{aligned}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msubsup> <mml:mo>∫</mml:mo> <mml:mn>1</mml:mn> <mml:mi>∞</mml:mi> </mml:msubsup> <mml:mfrac> <mml:mrow> <mml:mtext>d</mml:mtext> <mml:mi>y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that $$\textrm{P}\{ u(t,x)=\infty \quad \hbox { for all } t&gt;0 \hbox { and } x\in {\mathbb {R}}\}=1.$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>P</mml:mtext> <mml:mo>{</mml:mo> <mml:mi>u</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>∞</mml:mi> <mml:mspace/> <mml:mspace/> <mml:mtext>for all</mml:mtext> <mml:mspace/> <mml:mi>t</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> <mml:mspace/> <mml:mtext>and</mml:mtext> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> <mml:mo>}</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), 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. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022).

  PublisherOA PDFDOI

Recent grants

• Geometry of Random Fields and Stochastic Partial Differential Equations

  NSF · $411k · 2010–2014

• Collaborative Research: Asymptotic Geometry and Analysis of Stochastic Partial Differential Equations

  NSF · $255k · 2019–2023

• Collaborative Research: Fractals, Multifractals, and Stochastic Partial Differential Equations

  NSF · $211k · 2016–2019

• Random Fields and Stochastic Partial Differential Equations

  NSF · $415k · 2007–2011

• New Perspectives on Random Fields with Applications

  NSF · $295k · 2004–2008

Frequent coauthors

• Yimin Xiao

  Michigan State University

  54 shared

• Robert C. Dalang

  École Polytechnique Fédérale de Lausanne

  30 shared

• Kunwoo Kim

  30 shared

• Zhan Shi

  28 shared

• Eulàlia Nualart

  Pompeu Fabra University

  28 shared

• Daniel Conus

  25 shared

• Mathew Joseph

  20 shared

• Le Chen

  Auburn University

  19 shared

Labs

• Stochastics GroupPI