Related papers: Existence and space-time regularity for stochastic…

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…

This paper develops a fractional stochastic partial differential equation (SPDE) to model the evolution of a random tangent vector field on the unit sphere. The SPDE is governed by a fractional diffusion operator to model the L\'{e}vy-type…

We study the second-order quasi-linear stochastic partial differential equations (SPDEs) defined on $C^1$ domains. The coefficients are random functions depending on $t,x$ and the unknown solutions. We prove the uniqueness and existence of…

We consider the Stochastic Heat Equation (SHE) in $(1+1)$ dimensions with delta Dirac initial data and spacetime white noise. We prove exact large-time asymptotics for multi-point correlations of the SHE for strictly sublinear space…

Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…

This book is an introduction to the theory of stochastic partial differential equations (SPDEs), using the random field approach pioneered by J.B. Walsh (1986). It consists of two blocks: the core matter (Chapters 1 to 6) and the appendices…

In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport…

We consider a class of space-time coupled evolution equations (CEEs), obtained by a subordination of the heat operator. Our CEEs reformulate and extend known governing equations of non-Markovian processes arising as scaling limits of…

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index…

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the…

In the recent years there has been an increased interest in studying regularity properties of the derivatives of stochastic evolution equations (SEEs) with respect to their initial values. In particular, in the scientific literature it has…

We study a kind of better recurrence than Kolmogorov's one: periodicity recurrence,which corresponds periodic solutions in distribution for stochastic differential equations. On the basis of technique of upper and lower solutions and…

For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space.…

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a L\'evy process $X$ taking value in…

In this article, we consider the stochastic heat equation $du=(\Delta u+f(t,x))dt+ \sum_{k=1}^{\infty} g^{k}(t,x) \delta \beta_t^k, t \in [0,T]$, with random coefficients $f$ and $g^k$, driven by a sequence $(\beta^k)_k$ of i.i.d.…

We discuss some aspects of numerical continuation and bifurcation for partial differential equations, specifically pattern formation and coherent structures. For the sake of clarity we focus on wavetrains, stability and associated invasion…

The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties…

Realistic physical phenomena exhibit random fluctuations across many scales in the input and output processes. Models of these phenomena require stochastic PDEs. For three-dimensional coupled (vector-valued) stochastic PDEs (SPDEs), for…

We consider the stochastic heat equation (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero: \begin{equation*} \frac{\partial…

Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the past two decades, a growing interest has been devoted to stochastic finite element method (SFEM) for the propagation of…