Related papers: A stochastic heat equation with non-locally Lipsch…

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and…

Motivated by the regularization by noise phenomenon for SDEs we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation $$\frac{\partial u}{\partial t}=\frac12\frac{\partial^2 u}{\partial z^2}…

For every $R>0$, consider the stochastic heat equation $\partial_{t} u_{R}(t\,,x)=\tfrac12 \Delta_{S_{R}^{2}}u_{R}(t\,,x)+\sigma(u_{R}(t\,,x)) \xi_{R}(t\,,x)$ on $S_{R}^{2}$, where $\xi_{R}=\dot{W_{R}}$ are centered Gaussian noises with the…

We study the space-time nonlinear fractional stochastic heat equation driven by a space-time white noise, \begin{align*} \partial_t^\beta u(t,x)=-(-\Delta)^{\alpha/2}u(t,x)+I_t^{1-\beta}\Big[\sigma(u(t,x))\dot{W}(t,x)\Big],\ \ t>0, \ x\in…

Consider the nonlinear stochastic heat equation $$ \frac{\partial u (t,x)}{\partial t}=\frac{\partial^2 u (t,x)}{\partial x^2}+ \sigma(u (t,x))\dot{W}(t,x),\quad t> 0,\, x\in \mathbb{R}, $$ where $\dot W$ is a Gaussian noise which is white…

Consider the following nonlinear one-dimensional stochastic fractional heat equation $$\frac{\partial }{\partial t}u(t, x)= -(-\Delta)^{\alpha/2}u(t, x) +\sigma(t,x,u(t,x)) \dot{W}(t, x), $$ where $-(-\Delta)^{\alpha/2}$ is the fractional…

We put forward a new method for proving weak uniqueness of stochastic equations with singular drifts driven by a non-Markov or infinite-dimensional noise. We apply our method to study stochastic heat equation (SHE) driven by Gaussian…

We consider the stochastic heat equation on a compact smooth Riemannian manifold without boundary satisfying \begin{equation*} \partial_tu(t,x)=\frac{1}{2}\Delta_Mu(t,x)+\sigma(t,x,u)\dot{W}(t,x),\quad (t,x)\in\mathbb{R}_+\times M,…

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 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…

We study stochastic reaction--diffusion equation $$ \partial_tu_t(x)=\frac12 \partial^2_{xx}u_t(x)+b(u_t(x))+\dot{W}_{t}(x), \quad t>0,\, x\in D $$ where $b$ is a generalized function in the Besov space…

We consider a nonlinear stochastic heat equation on $[0,T]\times [-L,L]$, driven by a space-time white noise $W$, with a given initial condition $u_0: \mathbb{R} \to \mathbb{R}$ and three different types of (vanishing) boundary conditions:…

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 non-linear time-fractional stochastic heat type equation $$\frac{\partial^\beta u}{\partial t^\beta}+\nu(-\Delta)^{\alpha/2} u=I^{1-\beta}_t \bigg[\int_{\mathbb{R}^d}\sigma(u(t,x),h) \stackrel{\cdot}{\tilde N }(t,x,h)\bigg]$$…

Consider the $[0,1]$-valued continuous random field solution $(u_t(x))_{t\geq 0, x\in \mathbb R}$ to the one-dimensional stochastic heat equation \[ \partial_t u_t = \frac{1}{2}\Delta u_t + b(u_t) + \sqrt{u_t(1-u_t)} \dot W, \] where…

In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a L\'evy process $X$ taking value in…

Consider the following stochastic partial differential equation, \begin{equation*} \partial_t u_t(x)= \mathcal{L}u_t(x)+ \xi\sigma (u_t(x)) \dot F(t,x), \end{equation*} where $\xi$ is a positive parameter and $\sigma$ is a globally…

This paper studies the finite time explosion of the stochastic heat equation $\frac{\partial u}{\partial t}(t,x)=\frac{\partial^2}{\partial x^2} u(t,x)+(u(t,x))^{\beta}+\sigma(u(t,x))\dot{W}(t,x)$. We consider an interval $D=[-\pi,\pi]$…

For a class of non-linear stochastic heat equations driven by $\alpha$-stable white noises for $\alpha\in(1,2)$ with Lipschitz coefficients, we first show the existence and pathwise uniqueness of $L^p$-valued c\`{a}dl\`{a}g solutions to…

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…