Related papers: Spatial quadratic variations for the solution to a…
We deal with parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) with a small dispersion parameter based on high frequency data which are observed in time and space. By using the thinned…
This paper deals with the backward Euler method applied to semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive noise. The SPDE is discretized in space by the finite element method and in time by the…
We study the problem of learning the law of linear stochastic partial differential equations (SPDEs) with additive Gaussian forcing from spatiotemporal observations. Most existing deep learning approaches either assume access to the driving…
Fix $d\in\{1,2\}$, we consider a $d$-dimensional stochastic wave equation driven by a Gaussian noise, which is temporally white and colored in space such that the spatial correlation function is integrable and satisfies Dalang's condition.…
We introduce and test methods for the calibration of the diffusion term in Stochastic Partial Differential Equations (SPDEs) describing fluids. We take two approaches, one uses ideas from the singular value decomposition and the Biot-Savart…
We present a novel solution method for It\^o stochastic differential equations (SDEs). We subdivide the time interval into sub-intervals, then we use the quadratic polynomials for the approximation between two successive intervals. The main…
We study strong existence and pathwise uniqueness for stochastic differential equations in $\RR^d$ with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative…
In this article we investigate the numerical solution of a scalar semilinear stochastic delay differential equation (SDDE) where the linear instantaneous feedback and nonlinear delayed feedback terms are perturbed by a pair of standard…
The Stochastic Burgers Equation (SBE) is a singular, non-linear Stochastic Partial Differential Equation (SPDE) that describes, on mesoscopic scales, the fluctuations of stochastic driven diffusive systems with a conserved scalar quantity.…
The stochastic partial differential equation (SPDE) approach is widely used for modeling large spatial datasets. It is based on representing a Gaussian random field $u$ on $\mathbb{R}^d$ as the solution of an elliptic SPDE $L^\beta u =…
We prove the applicability of the Weighted Energy-Dissipation (WED) variational principle [50] to nonlinear parabolic stochastic partial differential equations in abstract form. The WED principle consists in the minimization of a…
We consider a 2D stochastic wave equation driven by a Gaussian noise, which is temporally white and spatially colored described by the Riesz kernel. Our first main result is the functional central limit theorem for the spatial average of…
An unsteady problem is considered for a space-fractional diffusion equation in a bounded domain. A first-order evolutionary equation containing a fractional power of an elliptic operator of second order is studied for general boundary…
In this paper, we study almost periodic solutions for semilinear stochastic differential equations driven by L\'{e}vy noise with exponential dichotomy property. Under suitable conditions on the coefficients, we obtain the existence and…
Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is an Hilbert-Schmidt perturbation of a fixed bounded…
We study stochastic partial differential equations (SPDEs) with potentially very rough fractional noise with Hurst parameter $H\in(0,1)$. Close to a change of stability measured with a small parameter $\varepsilon$, we rely on the natural…
The existence of random attractors for singular stochastic partial differential equations (SPDE) perturbed by general additive noise is proven. The drift is assumed only to satisfy the standard assumptions of the variational approach to…
This paper studies formulations of second-order elliptic partial differential equations in nondivergence form on convex domains as equivalent variational problems. The first formulation is that of Smears \& S\"uli [SIAM J.\ Numer.\ Anal.\…
Unlike many deterministic PDEs, stochastic equations are not amenable to the classical variational theory of Euler-Lagrange. In this paper, we show how self-dual variational calculus leads to solutions of various stochastic partial…
It has recently been shown that the evolution of a linear Partial Differential Equation (PDE) can be more conveniently represented in terms of the evolution of a higher spatial derivative of the state. This higher spatial derivative (termed…