Related papers: Quadratic covariations for the solution to a stoch…

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance \int_0^{s\wedge t} u^a [(t-u)^b+(s-u)^b]du, parameters a>-1, -1<b\leq 1, |b|\leq 1+a, corresponds to fractional Brownian…

We consider a stochastic process $Y$ defined by an integral in quadratic mean of a deterministic function $f$ with respect to a Gaussian process $X$, which need not have stationary increments. For a class of Gaussian processes $X$, it is…

We study the properties of a stochastic heat equation with a generalized mixed fractional Brownian noise. We obtain the covariance structure, stationarity and obtain bounds for the asymptotic behaviour of the solution. We suggest estimators…

Consider the stochastic heat equation \begin{equation*} \partial_t u_t(x)=\frac12 \partial^2_{xx}u_t(x) +b(u_t(x))+\dot{W}_{t}(x),\quad t\in(0,T],\, x\in D, \end{equation*} where $b$ is a generalized function, $D$ is either $[0,1]$ or…

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…

We establish the strong comparison principle and strict positivity of solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x))…

This paper deals with the long term behavior of the solution to the nonlinear stochastic heat equation $\partial u /\partial t - \frac{1}{2}\Delta u = b(u)\dot{W}$, where $b$ is assumed to be a globally Lipschitz continuous function and the…

In these lecture notes, we explore the mathematical preliminaries and foundational concepts that connect stochastic processes with partial differential equations. We begin by investigating Brownian motion, which serves as a model for random…

We consider fractional stochastic heat equations of the form $\frac{\partial u_t(x)}{\partial t} = -(-\Delta)^{\alpha/2} u_t(x)+\lambda \sigma (u_t(x)) \dot F(t,\, x)$. Here $\dot F$ denotes the noise term. Under suitable assumptions, we…

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:…

We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques…

In this note we focus our attention on a stochastic heat equation defined on the Heisenberg group $\mathbf{H}^{n}$ of order $n$. This equation is written as $\partial_t u=\frac{1}{2}\Delta u+u\dot{W}_\alpha$, where $\Delta$ is the…

Let $u(t, x) = (u_1(t, x), \dots, u_d(t, x))$ be the solution to the systems of nonlinear stochastic heat equations \[ \begin{split} \frac{\partial}{\partial t} u(t, x) &= \frac{\partial^2}{\partial x^2} u(t, x) + \sigma(u(t, x)) \dot{W}(t,…

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $\alpha \in (0,2)$. We consider…

Let $B^{a,b}$ be a weighted fractional Brownian motion with indices $a,b$ satisfying $a>-1,-1<b<0,|b|<1+a$. In this paper, motivated by the asymptotic property $$ E[(B^{a,b}_{s+\varepsilon}-B^{a,b}_s)^2] =O(\varepsilon^{1+b})\not\sim…

The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv…

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…

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial…

We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\left(\partial^\alpha - \textit{B} \right) u(t,x)= u(t,x) \cdot \dot{W}(t,x),$$ where $$\alpha \in (1/2, 1)\cup(1, 2);$$ $\textit{B}$…

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…