Mohammud Foondun

On the stochastic heat equation with spatially-colored random forcing

We consider the stochastic heat equation of the following form ∂ ∂t ut(x) = (Lut)(x) + b(ut(x)) + σ(ut(x))Ḟt(x) for t > 0, x ∈ R, where L is the generator of a Levy process and Ḟ is a spatially-colored, temporally white, gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u0 is a bounded and measurable function and σ is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case that Lu is replaced by its massive/dispersive analogue Lu − λu where λ ∈ R. And we describe accurately the effect of the parameter λ on the intermittence of the solution in the case that σ(u) is proportional to u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is the case when σ is identically equal to one [any other constant works also]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

A local-time correspondence for stochastic partial differential equations

It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the “spatial operator” is the L-generator of a Levy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly bigger than one. In addition, we prove that the solution to the SPDE is [Holder] continuous in its spatial variable if and only if the said local time is [Holder] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear L-space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of X, on the other hand. We study mainly linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a random-field solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Holder] continuous if and only if the solution to the nonlinear equation is. And the solutions are bounded and unbounded together as well. Finally, we prove that in the cases that the solutions are unbounded, they almost surely blow up at exactly the same points. Date: October 7, 2007. 2000 Mathematics Subject Classification. Primary. 60H15, 60J55; Secondary. 35R60, 35D05.

On the global maximum of the solution to a stochastic heat equation with compact-support initial data

Consider a stochastic heat equation @tu = @ 2 xxu + (u) _ w for a spacetime white noise _ w and a constant > 0. Under some suitable conditions on the initial function u0 and , we show that the quantities

Moment bounds for a class of fractional stochastic heat equations

We consider fractional stochastic heat equations of the form

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

\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)

Corrections and improvements to: “On the stochastic heat equation with spatially-colored random forcing”

. Here

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

\dot F

Intermittence and nonlinear parabolic stochastic partial differential equations

denotes the noise term. Under suitable assumptions, we show that the second moment of the solution grows exponentially with time. In particular, this answers an open problem in \cite{CoKh}. Along the way, we prove a number of other interesting properties which extend and complement results in \cite{foonjose}, \cite{Khoshnevisan:2013aa} and \cite{Khoshnevisan:2013ab}.

Heat kernel estimates and Harnack inequalities for some Dirichlet forms with non-local part

Abstract In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: ∂tβut(x)=−ν(−Δ)α/2ut(x)+It1−β[λσ(u)F⋅(t,x)]