Related papers: Stochastic comparisons for stochastic heat equatio…
Using tools from the theory of random fields with stationary increments, we introduce a new class of processes which can be used as a model for the noise perturbing an SPDE. This type of noise (called harmonizable) is not necessarily…
We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $\lambda\beta e^{\beta u }$, forced by an additive space-time white noise. We…
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…
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 study the effect of curvature on the Parabolic Anderson model by posing it over a Cartan-Hadamard manifold. We first construct a family of noises white in time and colored in space parameterized by a regularity parameter $\alpha$, which…
We consider the stochastic heat equation with multiplicative white noise: $\partial_t u =\partial_x^2u + b(u) +\sigma(u) \dot W$, both on $[0,1]$ and $\mathbf{R}$. In the case of $[0,1]$ we show that the finite Osgood criterion on $b$ is a…
We consider the one-dimensional stochastic heat and wave equations driven by Gaussian noises with constant initial conditions. We study the spatial average of the solutions on an interval of length $R$ and show that the family of laws of…
We investigate the strict positivity and the compact support property of solutions to the one-dimensional nonlinear stochastic heat equation: $$\partial_t u(t,x) = \frac{1}{2}\partial^2_x u(t,x) + \sigma(u(t,x))\dot{W}(t,x), \quad (t,x)\in…
In light of recent work on particles fluctuating in linear viscoelastic fluids, we study a linear stochastic partial-integro-differential equation with memory that is driven by a stationary noise on a bounded, smooth domain. Using the…
We consider a system of d non-linear stochastic heat equations in spatial dimension 1 driven by d-dimensional space-time white noise. The non-linearities appear both as additive drift terms and as multipliers of the noise. Using techniques…
We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure $\mu$…
This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where…
We study the linear stochastic fractional heat equation $$ \frac{\partial}{\partial t}u(t,x)=-(-\Delta)^{\frac{\alpha}2}u (t,x)+\dot{W}(t,x),\ \ t> 0,\ \ x\in\RR, $$ where $-(-\Delta)^{\frac{\alpha}{2}}$ denotes the fractional Laplacian…
We prove the existence and uniqueness of mild solution for the stochastic partial differential equation $$\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);$$ $\textit{B}$…
In this paper we study the Large Deviation Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space $\mathbb{R}^d$, with arbitrary dimension $d\geq 1$, under random influence which…
Consider the stochastic partial differential equation $\partial_t u = Lu+\sigma(u)\xi$, where $\xi$ denotes space-time white noise and $L:=-(-\Delta)^{\alpha/2}$ denotes the fractional Laplace operator of index…
We use the theory of regularity structures to develop an It\^o formula for $u$, the solution of the one dimensional stochastic heat equation driven by space-time white noise with periodic boundary conditions. In particular for any smooth…
We prove the existence of a sticky-reflected solution to the heat equation on the spatial interval $[0,1]$ driven by colored noise. The process can be interpreted as an infinite-dimensional analog of the sticky-reflected Brownian motion on…
In this paper we consider a system of non-linear stochastic heat equations on $\mathbb{R}^d$ driven by a Gaussian noise which is white in time and has a homogeneous spatial covariance. Under some suitable regularity and non degeneracy…
A study of a non-linear parabolic SPDEs of the form $\partial_{t}u=\mathcal{L}\,u + \sigma(u)f(B_t^x,t)\dot{w}$ with $\dot{w}$ as the space-time white noise and $f(B_t^x,t)$ a space-time harmonic function was done. The function…