Related papers: A Note on a Fenyman-Kac-Type Formula

We consider the (unique) mild solution $u(t,x)$ of a 1-dimensional stochastic heat equation on $[0,T]\times\mathbb R$ driven by time-homogeneous white noise in the Wick-Skorokhod sense. The main result of this paper is the computation of…

We consider the fractional stochastic heat equation on the $d$-dimensional torus $\mathbb{T}^d:=\left[-\frac{1}{2},\frac{1}{2}\right]^d$, $d\geq 1$, with periodic boundary conditions: \[ \partial_t u(t,\textbf{x})=…

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence…

We study the solution to a nonlinear stochastic heat equation in $d\geq 3$. The equation is driven by a Gaussian multiplicative noise that is white in time and smooth in space. For a small coupling constant, we prove (i) the solution…

In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussion noise with Hurst index $H\in(\frac{1}{2},1)$. A sharp regularity estimate of the mild solution and the numerical…

We study a time-fractional stochastic heat inclusion driven by additive time-space Brownian and L\'evy white noise. The fractional time derivative is interpreted as the Caputo derivative of order $\alpha \in (0,2).$ We show the following:…

In [HHL+17] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance…

In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in…

We consider the stochastic fractional heat equation $\partial_{t}u=\triangle^{\alpha/2}u+\lambda\sigma(u)\dot{w}$ on $[0,L]$ with Dirichlet boundary conditions, where $\dot{w}$ denotes the space-time white noise. For any $\lambda>0$, we…

Consider the solution $\mathcal{Z}(t,x)$ of the one-dimensional stochastic heat equation, with a multiplicative spacetime white noise, and with the delta initial data $\mathcal{Z}(0,x) = \delta(x)$. For any real $p>0$, we obtained detailed…

In this paper, we consider a quasi-linear stochastic heat equation on $[0,1]$, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs…

We derive an It\^o's-type formula for the one dimensional stochastic heat equation driven by a space-time white noise. The proof is based on elementary properties of the $\mathcal{S}$-transform and on the explicit representation of the…

We use a version of the Skorokhod integral to give a simple and rigorous formulation of the Wick-ordered (stochastic) heat equation with planar white noise, representing the free energy of an undirected random polymer. The solution for all…

In this project we investigate the stochastic Burgers' equation with multiplicative space-time white noise on an unbounded spatial domain. We give a random field solution to this equation by defining a process via a kind of Feynman-Kac…

In this paper, we study the stochastic heat equation driven by a multiplicative space-time $G$-white noise within the framework of sublinear expectations. The existence and uniqueness of the mild solution are proved. By generalizing the…

We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation…

We approximate the white-noise driven stochastic heat equation by replacing the fractional Laplacian by the generator of a discrete time random walk on the one dimensional lattice, and approximating white noise by a collection of i.i.d.…

We give sharp regularity results for the solution to the stochastic wave equation with linear fractional-colored noise. We apply these results in order to establish upper and lower bound for the hitting probabilities of the solution in…

For the stochastic partial differential equation $\frac{\partial u}{\partial t}=\mathcal L u +u\dot W$ where $\dot W$ is Gaussian noise colored in time and $\mathcal L$ is the infinitesimal generator of a Feller process $X$, we obtain…

Let $\{u(t,x)\}_{t>0,x\in{{\mathbb R}^{d}}}$ denote the solution to the linear (fractional) stochastic heat equation. We establish rates of convergence with respect to the uniform distance between the density of spatial averages of solution…