Fractional Stochastic Heat Equation Research Articles

A note on error estimation for hypothesis testing problems for some linear SPDEs

The aim of the present paper is to estimate and control the Type I and Type II errors of a simple hypothesis testing problem of the drift/viscosity coefficient for stochastic fractional heat equation driven by additive noise. Assuming that one path of the first $$N$$ Fourier modes of the solution is observed continuously over a finite time interval $$[0,T]$$ , we propose a new class of rejection regions and provide computable thresholds for $$T$$ , and $$N$$ , that guarantee that the statistical errors are smaller than a given upper bound. The considered tests are of likelihood ratio type. The main ideas, and the proofs, are based on sharp large deviation bounds. Finally, we illustrate the theoretical results by numerical simulations.

Stochastic delay fractional evolution equations driven by fractional Brownian motion

In this paper, we consider a class of stochastic delay fractional evolution equations driven by fractional Brownian motion in a Hilbert space. Sufficient conditions for the existence and uniqueness of mild solutions are obtained. An application to the stochastic fractional heat equation is presented to illustrate the theory.

Weak Convergence for a Class of Stochastic Fractional Equations Driven by Fractional Noise

We consider a class of stochastic fractional equations driven by fractional noise ont,x∈0,T×0,1 ∂u/∂t=Dδαu+ft,x,u+∂2BHt,x/∂t ∂x, with Dirichlet boundary conditions. We formally replace the random perturbation by a family of sequences based on Kac-Stroock processes in the plane, which approximate the fractional noise in some sense. Under some conditions, we show that the real-valued mild solution of the stochastic fractional heat equation perturbed by this family of noises converges in law, in the space𝒞0,T×0,1of continuous functions, to the solution of the stochastic fractional heat equation driven by fractional noise.

Additive Functionals of the Solution to Fractional Stochastic Heat Equation

We give sharp regularity results for the solution to the linear stochastic heat equation with fractional noise in time. We apply these result to prove the local nondeterminism of this process and to study the existence and the joint continuity of its local times.

Chung’s law of the iterated logarithm for anisotropic Gaussian random fields

In this paper we establish Chung’s law of the iterated logarithm for a class of anisotropic Gaussian random fields with stationary increments. This result is applicable to space–time Gaussian random fields and solution to the stochastic fractional heat equation.

The fractional stochastic heat equation on the circle: Time regularity and potential theory

We consider a system of d linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle S 1 . We obtain sharp results on the Hölder continuity in time of the paths of the solution u = { u ( t , x ) } t ∈ R + , x ∈ S 1 . We then establish upper and lower bounds on hitting probabilities of u , in terms of the Hausdorff measure and Newtonian capacity respectively.

Spatiotemporal random fields associated with stochastic fractional Helmholtz and heat equations

In this paper, a class of spatiotemporal random field models defined as mean-square solutions of fractional versions of the stochastic heat equation are considered. Different sampling schemes in space and time are introduced to solve the problem of estimation of fully parameterized spatiotemporal random fields.

Time regularity of the evolution solution to fractional stochastic heat equation

We study the time-regularity of the paths of solutions to stochastic partial differential equations (SPDE) driven by additive infinite-dimensional fractional Brownian noise. Sharp sufficient conditions for almost-sure Holder continuity, and other, more irregular levels of uniform continuity, are given when the space parameter is fixed. Additionally, a result is included on time-continuity when the solution is understood as a spatially Holder-continuous-function-valued stochastic process. Tools used for the study include the Brownian representation of fractional Brownian motion, and sharp Gaussian regularity results.

Fractional random fields associated with stochastic fractional heat equations

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

Fractional random fields associated with stochastic fractional heat equations

This paper introduces a convenient class of spatiotemporal random field models that can be interpreted as the mean-square solutions of stochastic fractional evolution equations.

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.