Related papers: Weak nonmild solutions to some SPDEs
In [HHL+17] the authors showed existence and uniqueness of solutions to the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and rougher than white in space (in particular, its covariance…
We consider a class of stochastic heat equations driven by truncated $\alpha$-stable white noises for $1<\alpha<2$ with noise coefficients that are continuous but not necessarily Lipschitz and satisfy globally linear growth conditions. We…
In this paper, we study the stochastic heat equation driven by a multiplicative space-time $G$-white noise within the framework of sublinear expectations. The existence and uniqueness of the mild solution are proved. By generalizing the…
We consider a family of nonlinear stochastic heat equations of the form $\partial_t u=\mathcal{L}u + \sigma(u)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $\mathcal{L}$ the generator of a symmetric L\'evy process on $\R$, and…
In this paper, we consider a quasi-linear stochastic heat equation on $[0,1]$, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs…
We investigate a stochastic partial differential equation with second order elliptic operator in divergence form, having a piecewise constant diffusion coefficient, and driven by a space-time white noise. We introduce a notion of weak…
We find the weak rate of convergence of the spatially semidiscrete finite element approximation of the nonlinear stochastic heat equation. Both multiplicative and additive noise is considered under different assumptions. This extends an…
We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric L\'evy white noise. We identify conditions for existence for these two kinds of solutions,…
This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textless{}H\textless{}1/2 in…
We apply the well-known Banach-Necas-Babuska inf-sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the the data and on…
We consider a stochastic partial differential equation with piecewise constant coefficients driven by a multiplicative space-time white noise. The existence and uniqueness of the mild solution in Walsh sense is established. We mainly study…
We consider a stochastic heat equation driven by a space-time white noise and with a singular drift, where a local-time in space appears. The process we study has an explicit invariant measure of Gibbs type, with a non-convex potential. We…
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:…
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 L\'evy process in the plane and we use it to construct a family of complex-valued random fields that we show to converge in law, in the space of continuous functions, to a complex Brownian sheet. We apply this result to obtain…
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 (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero: \begin{equation*} \frac{\partial…
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…
Consider the non-linear stochastic fractional-diffusion equation \begin{eqnarray*} \left \{\begin{array}{lll} \frac{\partial}{\partial t}u(x,t)= -( \Delta)^{\alpha/2} u^m(x,t) + \sigma(u(x,t)) \dot{W}(x,t),\, x\in \mathbb{R}^d,t>0, u(x,0)=…
In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index…