Class Of Stochastic Heat Equations Research Articles

Existence of weak solutions to stochastic heat equations driven by truncated α-stable white noises with non-Lipschitz coefficients

We consider a class of stochastic heat equations driven by truncated α-stable white noises for α∈(1,2) with noise coefficients that are continuous but not necessarily Lipschitz continuous. We prove the existence of weak solution in probabilistic sense, taking values in two different forms under different conditions, to such an equation using a weak convergence argument on solutions to the approximating stochastic heat equations. More precisely, for α∈(1,2) there exists a measure-valued weak solution. However, for α∈(1,5/3) there exists a function-valued weak solution, and in this case we further show that for p∈(α,5/3) the uniform p-th moment in Lp-norm of the weak solution is finite, and that the weak solution is uniformly stochastic continuous in Lp sense.

Chung’s law of the iterated logarithm for a class of stochastic heat equations

Chung’s law of the iterated logarithm for a class of stochastic heat equations

Moment bounds of a class of stochastic heat equations driven by space–time colored noise in bounded domains

We consider the fractional stochastic heat type equation ∂∂tut(x)=−(−Δ)α∕2ut(x)+ξσ(ut(x))Ḟ(t,x),x∈D,t>0,with nonnegative bounded initial condition, where α∈(0,2], ξ>0 is the noise level, σ:R→R is a globally Lipschitz function satisfying some growth conditions and the noise term Ḟ behaves in space like the Riez kernel and is possibly correlated in time and D is the unit open ball centered at the origin in Rd. When the noise term is not correlated in time, we establish a change in the growth of the solution of these equations depending on the noise level ξ. On the other hand when the noise term behaves in time like the fractional Brownian motion with index H∈(1∕2,1), We also derive explicit bounds leading to a well-known intermittency property.

A non-uniform discretization of stochastic heat equations with multiplicative noise on the unit sphere

We investigate a discretization of a class of stochastic heat equations on the unit sphere with multiplicative noise. A spectral method is used for the spatial discretization and the truncation of the Wiener process, while an implicit Euler scheme with non-uniform steps is used for the temporal discretization. Some numerical experiments inspired by Earth’s surface temperature data analysis GISTEMP provided by NASA are given.

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 works of [Stoch. Stoch. Rep. 37 (1991) 225–245] and [Ann. Probab. 45 (2017) 377–403] among others. 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.

On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian

Consider stochastic heat equations with fractional Laplacian on Rd. The driving noise is generalized Gaussian which is white in time but spatially homogeneous. We study the large-scale structure of the tall peaks for (i) the linear stochastic heat equation and (ii) the parabolic Anderson model. We obtain the largest order of the peaks and compute the macroscopic Hausdorff dimensions of the peaks for (i) and (ii). These result imply that both (i) and (ii) exhibit multi-fractal behavior even though only (ii) is intermittent. This is an extension of a result of Khoshnevisan et al. (2017) to a wider class of stochastic heat equations.

On non-existence of global solutions to a class of stochastic heat equations

We consider nonlinear parabolic SPDEs of the form ! tu = ! (! ! )! /2u + b(u )+ (u )˙ w ,w here ˙ w denotes space-time white noise. The functions b and are both locally Lipschitz continuous. Under some suitable conditions on the parameters of this SPDE, we show that the above equation has no random-field solution. This complements recent works of Khoshnevisan and his coauthors.

Large deviation principle for stochastic heat equation with memory

In this work, using the weak convergence argument, we prove a Freidlin-Wentzell's large deviation principle for a class of stochastic heat equations with memory and Dirichlet boundary conditions, where the nonlinear term is allowed to be of polynomial growth.

On a stochastic heat equation with first order fractional noises and applications to finance

In this paper, we propose a class of stochastic heat equations with first order fractional noises. We define a first order noise through the adjoint operator of the first order operator, where the operation of the stochastic integral can be avoided. In this framework, the existence and uniqueness of the solution of the equation will be established. Further, we give the regularity of the solution. Finally, we model the term structure of forward rate with the solutions and give the conditions under which the market is arbitrary-free.

Tail estimates for positive solutions of stochastic heat equation

In this paper, we study the solution of a class of stochastic heat equations of convolution type. We give an explicit solution X t using two basic tools: the characterization theorem for generalized functions and the convolution calculus. For positive initial condition f and coefficients processes Vt, Mt, we prove that the corresponding solution X t admits an integral representation by a certain measure. Finally, we compute the tail estimate for the obtained solution and its expectation.