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Mohammud Foondun

Publications and source records attributed to Mohammud Foondun.

At least 19 recordsLinked to original sources
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.

math.PR↗
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.

math.PR↗
arXiv2023

Global solution for superlinear stochastic heat equation on $\mathbb{R}^d$ under Osgood-type conditions

We study the \textit{stochastic heat equation} (SHE) on $\R^d$ subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists random field solution which do not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations.

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].

math.PR↗
arXiv2022

Small ball probability estimates for the Hölder semi-norm of the stochastic heat equation

We consider the stochastic heat equation on $[0,\,1]$ with periodic boundary conditions and driven by space-time white noise. Under various natural conditions, we study small ball probabilities for the Hölder semi-norms of the solutions, and provide near optimal bounds on these probabilities. As an application, we prove a support theorem in these Hölder semi-norms.

math.PR↗
arXiv2021

An inequality for a class of Markov processes

Let $α\in (0,2)$ and consider the operator $\sL$ given by \[ \sL f(x)=\int[ f(x+h)-f(x)-1_{(|h|\leq 1)}h\cdot \grad f(x)]\frac{n(x,h)}{|h|^{d+α}} \d h, \] where the term $1_{(|h|\leq 1)}h\cdot \grad f(x)$ is not present when $α\in (0,1)$. Under some suitable assumptions on the kernel $n(x,h)$, we prove a Krylov-type inequality for processes associated with $\sL$. As an application of the inequality, we prove the existence of a solution to the martingale problem for $\sL$ without assuming any continuity of $n(x,h)$.

math.PR↗
arXiv2020

The Osgood condition for stochastic partial differential equations

We study the following equation \begin{equation*} \frac{\partial u(t,\,x)}{\partial t}= Δu(t,\,x)+b(u(t,\,x))+σ\dot{W}(t,\,x),\quad t>0, \end{equation*} where $σ$ is a positive constant and $\dot{W}$ is a space-time white noise. The initial condition $u(0,x)=u_0(x)$ is assumed to be a nonnegative and continuous function. We first study the problem on $[0,\,1]$ with homogeneous Dirichlet boundary conditions. Under some suitable conditions, together with a theorem of Bonder and Groisman, our first result shows that the solution blows up in finite time if and only if \begin{equation*} \int_{\cdot}^\infty\frac{1}{b(s)}\,d s<\infty, \end{equation*} which is the well-known Osgood condition. We also consider the same equation on the whole line and show that the above condition is sufficient for the nonexistence of global solutions. Various other extensions are provided; we look at equations with fractional Laplacian and spatial colored noise in $\mathbb{R}^d$.

math.PR↗
arXiv2019

Spatial asymptotics and strong comparison principle for some fractional stochastic heat equations

Consider the following stochastic heat equation, \begin{align*} \frac{\partial u_t(x)}{\partial t}=-ν(-Δ)^{α/2} u_t(x)+σ(u_t(x))\dot{F}(t,\,x), \quad t>0, \; x \in R^d. \end{align*} Here $-ν(-Δ)^{α/2}$ is the fractional Laplacian with $ν>0$ and $α\in (0,2]$, $σ: R\rightarrow R$ is a globally Lipschitz function, and $\dot{F}(t,\,x)$ is a Gaussian noise which is white in time and colored in space. Under some suitable additional conditions, we prove a strong comparison theorem and explore the effect of the initial data on the spatial asymptotic properties of the solution. This constitutes an important extension over a series of recent works.

math.PR↗
arXiv2018

Remarks on a fractional-time stochastic equation

We consider a class of fractional time stochastic equation defined on a bounded domain and show that the presence of the time derivative induces a significant change in the qualitative behaviour of the solutions. This is in sharp contrast with the phenomenon showcased in earlier works. We also show that as one {\it tunes off the fractional} in the fractional time derivative, the solution behaves more and more like its {\it usual} counterpart.

math.PR↗
arXiv2018

Critical parameters for reaction-diffusion equations involving space-time fractional derivatives

We will look at reaction-diffusion type equations of the following type, $$\partial^β_tV(t,x)=-(-Δ)^{α/2} V(t,x)+I^{1-β}_t[V(t,x)^{1+η}].$$ We first study the equation on the whole space by making sense of it via an integral equation. Roughly speaking, we will show that when $0<η\leqη_c$, there is no global solution other than the trivial one while for $η>η_c$, non-trivial global solutions do exist. We also study the equation on a bounded domain with Dirichlet boundary condition and show that the presence of the time derivative induces a significant change in the behaviour of the solution.

math.AP↗
arXiv2016

Some properties of non-linear fractional stochastic heat equations on bounded domains

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ ξσ(u_t(x)) \dot F(t,x), \end{equation*} where $ξ$ is a positive parameter and $σ$ is a globally Lipschitz continuous function. The stochastic forcing term $\dot F(t,x)$ is white in time but possibly colored in space. The operator $\mathcal{L}$ is a non-local operator. We study the behaviour of the solution with respect to the parameter $ξ$, extending the results in \cite{FoonNual} and \cite{Bin}

math.PR↗
arXiv2016

Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains

In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[λσ(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0, β\in (0,1)$, $α\in (0,2]$. The operator $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2} $ is the generator of an isotropic stable process and $I^{1-β}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity $σ:\RR{R}\to\RR{R}$ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane(J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326). We first study the existence and uniqueness of the solution of these equations {and} under suitable conditions on the initial function, we {also} study the asymptotic behavior of the solution with respect to the parameter $λ$. In particular, our results are significant extensions of those in Foondun et al (M. Foondun, K. Tian and W. Liu. On some properties of a class of fractional stochastic equations. Preprint available at arxiv.org 1404.6791v1.), Foondun and Khoshnevisan (M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568.), Nane and Mijena (J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326; J. B. Mijena, and E.Nane. Intermittence and time fractional partial differential equations. Submitted. 2014).

math.PR↗
arXiv2016

Some non-existence results for a class of stochastic partial differential equations

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ σ(u_t(x))\dot F(t,x)\quad{t>0}\quad\text{and}\quad x\in R^d. \end{equation*} The operator $\mathcal{L}$ is the generator of a strictly stable process and $\dot F$ is the random forcing term which is assumed to be Gaussian. Under some additional conditions, most notably on $σ$ and the initial condition, we show non-existence of global random field solutions. Our results are new and complement earlier works.

math.PR↗
arXiv2016

An approximation result for a class of stochastic heat equations with colored noise

We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier results. Among other things, our results enable us to obtain sharp estimates on the moments of the solution. A main technical ingredient of our method is a local limit theorem which is of independent interest.

math.PR↗
arXiv2015

Asymptotic properties of some space-time fractional stochastic equations

Consider non-linear time-fractional stochastic heat type equations of the following type, $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[λσ(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0, β\in (0,1)$, $α\in (0,2]$. The operator $\partial^β_t$ is the Caputo fractional derivative while $-(-Δ)^{α/2} $ is the generator of an isotropic stable process and $I^{1-β}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. And the multiplicative non-linearity $σ$ is assumed to be globally Lipschitz continuous. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter $λ$. In particular, our results are significant extensions of existing results. Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

math.PR↗
arXiv2014

On the behaviour of stochastic heat equations on bounded domains

Consider the following equation $$\partial_t u_t(x)=\frac{1}{2}\partial _{xx}u_t(x)+λσ(u_t(x))\dot{W}(t,\,x)$$ on an interval. Under Dirichlet boundary condition, we show that in the long run, the second moment of the solution grows exponentially fast if $λ$ is large enough. But if $λ$ is small, then the second moment eventually decays exponentially. If we replace the Dirichlet boundary condition by the Neumann one, then the second moment grows exponentially fast no matter what $λ$ is. We also provide various extensions.

math.PR↗