The stochastic wave equation with fractional noise: A random field approach

We consider the linear stochastic wave equation with spatially homogeneous Gaussian noise, which is fractional in time with index H>1/2$H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation of the condition obtained in Dalang (1999) [10], where the noise is white in time. Under this condition, we show that the solution is L²(Ω)$L^2\left(\Omega \right)$-continuous. Similar results are obtained for the heat equation. Unlike in the white noise case, the necessary and sufficient condition for the existence of the solution in the case of the heat equation is different (and more general) than the one obtained for the wave equation.     

Systems of stochastic partial differential equations with reflection: Existence and uniqueness

In this paper, we establish the existence and uniqueness of solutions of systems of stochastic partial differential equations (SPDEs) with reflection in a convex domain. The lack of comparison theorems for systems of SPDEs makes things delicate.     

Lower bound technique in the theory of a stochastic differential equation

We establish a criterion for the existence of an invariant measure for Markov processes acting on measures defined on an arbitrary complete separable metric space. This criterion is applied to time-homogeneous Markov processes associated with a nonlinear heat equation driven by an impulsive noise.     

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

We consider a system of d$d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle S¹$S^1$. We obtain sharp results on the Hölder continuity in time of the paths of the solution u=_{{u(t,x)}t∈R+,x∈S1}$u={\left\{u(t,x)\right\}}_{t\in {\mathbb{R}}_{+},x\in S^1}$. We then establish upper and lower bounds on hitting probabilities of u$u$, in terms of the Hausdorff measure and Newtonian capacity respectively.     

Large deviations for stochastic partial differential equations driven by a Poisson random measure

Stochastic partial differential equations driven by Poisson random measures (PRMs) have been proposed as models for many different physical systems, where they are viewed as a refinement of a corresponding noiseless partial differential equation (PDE). A systematic framework for the study of probabilities of deviations of the stochastic PDE from the deterministic PDE is through the theory of large deviations. The goal of this work is to develop the large deviation theory for small Poisson noise perturbations of a general class of deterministic infinite dimensional models. Although the analogous questions for finite dimensional systems have been well studied, there are currently no general results in the infinite dimensional setting. This is in part due to the fact that in this setting solutions may have little spatial regularity, and thus classical approximation methods for large deviation analysis become intractable. The approach taken here, which is based on a variational representation for nonnegative functionals of general PRMs, reduces the proof of the large deviation principle to establishing basic qualitative properties for controlled analogues of the underlying stochastic system. As an illustration of the general theory, we consider a particular system that models the spread of a pollutant in a waterway.     

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.     

On stochastic partial differential equations with Unbounded coefficients

Second-order stochastic partial differential equations of parabolic type are considered. Generalizations of known theorems on existence, uniqueness and on approximations are presented. Thus, in particular the case of unbounded coefficients in investigated. Some examples illustrating the usefulness of the results are also given.     

Semilinear ordinary differential equation coupled with distributed order fractional differential equation

The system , where D^γ,γ∈[0,2] are operators of fractional differentiation, is investigated and the existence of a mild and classical solution is proven. Also, a necessary and sufficient condition for the existence and uniqueness of a solution to a general linear fractional differential equation , in is given.     

Convergence of the increments of a stochastic integral associated to the stochastic wave equation

We study a type of one-dimensional wave equation on the plane with non-linear random forcing. We are interested in the almost sure behaviour of the normalized increments of the solution process associated to this type of wave equation. Also we study the behaviour of the normalized increments of some other stochastic integral equation.     

On quasi-linear stochastic partial differential equations

Summary We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.and INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR     

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

Consider stochastic heat equations with fractional Laplacian on ${\mathbb{R}}^d$. 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 uniqueness for nonlinear elliptic equation involving the Pucci's extremal operator

In this article we study uniqueness of positive solutions for the nonlinear uniformly elliptic equation in RN, limr→∞u(r)=0, where denotes the Pucci's extremal operator with parameters 0<λ?Λ and p>1. It is known that all positive solutions of this equation are radially symmetric with respect to a point in RN, so the problem reduces to the study of a radial version of this equation. However, this is still a nontrivial question even in the case of the Laplacian (λ=Λ). The Pucci's operator is a prototype of a nonlinear operator in no-divergence form. This feature makes the uniqueness question specially challenging, since two standard tools like Pohozaev identity and global integration by parts are no longer available. The corresponding equation involving is also considered.     

Skorohod integration and stochastic calculus beyond the fractional Brownian scale

We extend the Skorohod integral, allowing integration with respect to Gaussian processes that can be more irregular than any fractional Brownian motion. This is done by restricting the class of test random variables used to define Skorohod integrability. A detailed analysis of the size of this class is given; it is proved to be non-empty even for Gaussian processes which are not continuous on any closed interval. Despite the extreme irregularity of these stochastic integrators, the Skorohod integral is shown to be uniquely defined, and to be useful: an Ito formula is established; it is employed to derive a Tanaka formula for a corresponding local time; linear additive and multiplicative stochastic differential equations are solved; an analysis of existence for the stochastic heat equation is given.     

Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion

We study the approximation of stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H>1/2$H>1/2$. For the mean-square error at a single point we derive the optimal rate of convergence that can be achieved by arbitrary approximation methods that are based on an equidistant discretization of the driving fractional Brownian motion. We find that there are mainly two cases: either the solution can be approximated perfectly or the best possible rate of convergence is n^−H−1/2$n^{-H-1/2}$, where n$n$ denotes the number of evaluations of the fractional Brownian motion. In addition, we present an implementable approximation scheme that obtains the optimal rate of convergence in the latter case.     

Exponential stability for neutral stochastic partial differential equations with delays and Poisson jumps

In this paper, we consider a class of neutral stochastic partial differential equations with delays and Poisson jumps. Sufficient conditions for the existence and exponential stability in mean square as well as almost surely exponential stability of mild solutions are derived by means of the Banach fixed point principle. An example is provided to illustrate the effectiveness of the proposed result.     

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.