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Let $(\Omega, \mu)$ be a probability space endowed with an ergodic action, $\tau$ of $( {\mathbb R} ^n, +)$. Let $H(x,p; \omega)=H_\omega(x,p)$ be a smooth Hamiltonian on $T^* {\mathbb R} ^n$ parametrized by $\omega\in \Omega$ and such that…

Let $\{u(t\,,x)\}_{t\ge 0, x\in \mathbb{R}^d}$ denote the solution of a $d$-dimensional nonlinear stochastic heat equation that is driven by a Gaussian noise, white in time with a homogeneous spatial covariance that is a finite Borel…

We explore the relation between fast waves, damping and imposed noise for different scalings by considering the singularly perturbed stochastic nonlinear wave equations \nu u_{tt}+u_t=\D u+f(u)+\nu^\alpha\dot{W} on a bounded spatial domain.…

We study inhomogeneous heat equation with inverse square potential, namely, \[\partial_tu + \mathcal{L}_a u= \pm |\cdot|^{-b} |u|^{\alpha}u,\] where $\mathcal{L}_a=-\Delta + a |x|^{-2}.$ We establish some fixed-time decay estimate for…

Consider the stochastic heat equation $\partial_tu=\mathscr{L}u+\lambda\sigma(u)\xi$, where $\mathscr{L}$ denotes the generator of a L\'{e}vy process on a locally compact Hausdorff Abelian group $G$, $\sigma:\mathbf{R}\to\mathbf{R}$ is…

We study the nonlinear stochastic heat equation driven by space-time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We…

Let $\mathbb{H}^n$ be the $n$-dimensional real hyperbolic space, $\Delta$ its nonnegative Laplace--Beltrami operator whose bottom of the spectrum we denote by $\lambda_{0}$, and $\sigma \in (0,1)$. The aim of this paper is twofold. On the…

In this article we prove the continuity of the deterministic function $u:[0,T]\times \mathcal{\bar{D}}\rightarrow \mathbb{R}$, defined by $u(t,x):=Y_{t}^{t,x}$, where the process $(Y_{s}^{t,x})_{s\in[t,T]}$ is given by the generalized…

We consider the stochastic heat equation $$\frac{\partial Y_t(x)}{\partial t} = \frac{1}{2} \Delta_x Y_t(x) + Y_{t-}(x)^{\beta} \dot{L}^{\alpha}$$ with $t \ge 0$, $x \in \mathbb{R}$ and $L^{\alpha}$ being an $\alpha$-stable white noise…

In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coeffcient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this…

In this paper, we study the stochastic heat equation in the spatial domain $\mathbb{R}^d$ subject to a Gaussian noise which is white in time and colored in space. The spatial correlation can be any symmetric, nonnegative and…

We consider a $d$-dimensional random field $u = \{u(t,x)\}$ that solves a non-linear system of stochastic wave equations in spatial dimensions $k \in \{1,2,3\}$, driven by a spatially homogeneous Gaussian noise that is white in time. We…

We consider the stochastic heat equation on $\mathbb R^d$ with multiplicative space-time white noise noise smoothed in space. For $d\geq 3$ and small noise intensity, the solution is known to converge to a strictly positive random variable…

This article generalizes the small noise cutoff phenomenon to the strong solutions of the stochastic heat equation and the damped stochastic wave equation over a bounded domain subject to additive and multiplicative Wiener and L\'evy noises…

Let $p:\mathbb{C}\to\mathbb{R}$ be a subharmonic, nonharmonic polynomial and $\tau$ a real parameter. Define $\bar{Z}_{\tau p} = \partial_{\bar z} + \tau p_{\bar z}$, a closed, densely-defined operator on $L^2(\mathbb{C})$. If $\Box_{\tau…

We study the stochastic heat equation (SHE) $\partial_t u = \frac12 \Delta u + \beta u \xi$ driven by a multiplicative L\'evy noise $\xi$ with positive jumps and amplitude $\beta>0$, in arbitrary dimension $d\geq 1$. We prove the existence…

We consider a $d$-dimensional random field $u=(u(x), x\in D)$ that solves a system of elliptic stochastic equations on a bounded domain $D\subset \mathbb{R}^k$, with additive white noise and spatial dimension $k=1,2,3$. Properties of $u$…

In this paper we study the influence of the Hardy potential in the fractional heat equation. In particular, we consider the problem $$(P_\theta)\quad \left\{ \begin{array}{rcl} u_t+(-\Delta)^{s} u&=&\l\dfrac{\,u}{|x|^{2s}}+\theta u^p+ c…

We classify the smooth self-similar solutions of the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite…

We consider the generic divergence form second order parabolic equation with coefficients that are regular in the spatial variables and just measurable in time. We show that the spatial derivatives of its fundamental solution admit upper…