Related papers: Variations of the solution to a stochastic heat eq…

In this paper, we propose and analyze a new semi-implicit stochastic multiscale method for the radiative heat transfer problem with additive noise fluctuation in composite materials. In the proposed method, the strong nonlinearity term…

We consider a simple model of a stochastic heat engine, which consists of a single Brownian particle moving in a one-dimensional periodically breathing harmonic potential. Overdamped limit is assumed. Expressions of second moments…

We establish a unique continuation property for stochastic heat equations evolving in a bounded domain $G$. Our result shows that the value of the solution can be determined uniquely by means of its value on an arbitrary open subdomain of…

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like…

Because of the finiteness of the life span and boundedness of the physical space, the more reasonable or physical choice is the tempered power-law instead of pure power-law for the CTRW model in characterizing the waiting time and jump…

The Initial-Boundary Value Problem for the heat equation is solved by using a new algorithm based on a random walk on heat balls. Even if it represents a sophisticated generalization of the Walk on Spheres (WOS) algorithm introduced to…

We consider a nonlinear stochastic heat equation $\partial_tu=\frac{1}{2}\partial_{xx}u+\sigma(u)\partial_{xt}W$, where $\partial_{xt}W$ denotes space-time white noise and $\sigma:\mathbf {R}\to \mathbf {R}$ is Lipschitz continuous. We…

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional…

This paper studies the existence and uniqueness of solution of It\^o type stochastic differential equation $dx(t)=b(t, x(t), \om)dt+\si(t,x(t), \om) d B(t)$, where $B(t)$ is a fractional Brownian motion of Hurst parameter $H>1/2$ and…

We prove existence and uniqueness of the solution of a stochastic shell--model. The equation is driven by an infinite dimensional fractional Brownian--motion with Hurst--parameter $H\in (1/2,1)$, and contains a non--trivial coefficient in…

In this article we consider the stochastic heat equation $u_{t}-\Delta u=\dot B$ in $(0,T) \times \bR^d$, with vanishing initial conditions, driven by a Gaussian noise $\dot B$ which is fractional in time, with Hurst index $H \in (1/2,1)$,…

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an…

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations…

Consider the stochastic heat equation $\partial_t u = (\frac{\varkappa}{2})\Delta u+\sigma(u)\dot{F}$, where the solution $u:=u_t(x)$ is indexed by $(t,x)\in (0, \infty)\times\R^d$, and $\dot{F}$ is a centered Gaussian noise that is white…

In this note, we consider the stochastic heat and Schr\"odinger equation, and show that, at time $t$, the onset of the chaos occurs on the scale of $1/t$, and the Fourier spectrum of the solution is asymptotically Gaussian after centering…

We study the Gaussian fluctuations of a nonlinear stochastic heat equation in spatial dimension two. The equation is driven by a Gaussian multiplicative noise. The noise is white in time, smoothed in space at scale $\varepsilon$, and tuned…

In this paper we address again the problem of the connection between multitime Brownian sheet and heat type PDEs. The main results include: the volumetric character of the solutions of the forward (backward) diffusion-like PDEs; the forward…

This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $H\in(0,1/2)$. We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower…

We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy process on $\R$, and…

For the fractional heat equation $\frac{\partial}{\partial t} u(t,x) = -(-\Delta)^{\frac{\alpha}{2}}u(t,x)+ u(t,x)\dot W(t,x)$ where the covariance function of the Gaussian noise $\dot W$ is defined by the heat kernel, we establish…