Related papers: Space-time fractional SPDEs with locally Lipschitz…
Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad…
We introduce an explicit, adaptive time-stepping scheme for the simulation of SPDEs with one-sided Lipschitz drift coefficients. Strong convergence rates are proven for the full space-time discretisation with multiplicative trace-class…
This paper establishes a comprehensive well-posedness and regularity theory for time-fractional stochastic partial differential equations on $\mathbb{R}^d$ driven by mixed Wiener--L\'evy noises. The equations feature a Caputo time…
We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable:…
We study the following equation \begin{equation*} \frac{\partial u(t,\,x)}{\partial t}= \Delta u(t,\,x)+b(u(t,\,x))+\sigma \dot{W}(t,\,x),\quad t>0, \end{equation*} where $\sigma$ is a positive constant and $\dot{W}$ is a space-time white…
In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone…
We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of…
This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}\Delta^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions.…
We show that the initial value problem associated to the dispersive generalized Benjamin-Ono-Zakharov-Kuznetsov equation$$ u\_t-D\_x^\alpha u\_{x} + u\_{xyy} = uu\_x,\quad (t,x,y)\in\R^3,\quad 1\le \alpha\le 2,$$is locally well-posed in the…
We study the generalized derivative nonlinear Schr\"odinger equation $i\partial_t u+\Delta u = P(u,\overline{u},\partial_x u,\partial_x \overline{u})$, where $P$ is a polynomial, in Sobolev spaces. It turns out that when $\text{deg } P\geq…
We consider the periodic fractional nonlinear Schr\"{o}dinger equation $$ iu_t -(-\Delta)^{\frac{s}{2}} u + \mathcal{N}(|u|)u=0, \quad x\in \mathbb{T}^N,\, \, t \in \mathbb R, \, \, s>0, $$ where the nonlinearity term is expressed in two…
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 consider the following stochastic heat equation \begin{equation*} \partial_t u(t\,,x) = \tfrac12 \partial^2_x u(t\,,x) + b(u(t\,,x)) + \sigma(u(t\,,x)) \dot{W}(t\,,x), \end{equation*} defined for $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$,…
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains…
Covariant stochastic partial differential equations are studied in any dimension. A special class of such equations is selected and it is proven that the solutions can be analytically continued to Minkowski space-time yielding tempered…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise under more relaxed conditions. The SPDE is discretized…
We establish well-posedness results for multidimensional non degenerate $\alpha$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+\gamma}$ with $\gamma<1$ and $\alpha$ in $(1,2]$, where…
In this paper, we consider the KPZ equation driven by space-time white noise replaced with its fractional derivatives of order $\gamma>0$ in spatial variable. A well-posedness theory for the KPZ equation is established by Hairer [3] as an…
In this article, we consider the following class of stochastic partial differential equations (SPDE): \begin{equation*} \left\{\begin{aligned}\mathrm{d} \mathbf{X}(t)&=\mathrm{A}(t,\mathbf{X}(t))\mathrm{d}…
We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in…