相关论文: Analytically weak and mild solutions to stochastic…

In this article, we study the existence and uniqueness of a weak solution to the fractional single-phase lag heat equation. This model contains the terms $\cal{D}_t^\alpha(u_t)$ and $\cal{D}_t^\alpha u $ (with $\alpha \in(0,1)$), where…

We introduce and study a new class of non-Archimedean stochastic pseudodifferential equations. These equations are the non-Archimedean counterparts of the classical stochastic heat equations. We show the existence and uniqueness of mild…

We consider nonnegative solutions of the quasilinear heat equation $\partial_t u = \tfrac{1}{2} u \partial_x^2 u$ in one dimension. Our solutions may vanish and may be unbounded. The equation is then degenerate, and weak solutions are…

This paper aims at developing a systematic study for the weak rate of convergence of the Euler-Maruyama scheme for stochastic differential equations with very irregular drift and constant diffusion coefficients. We apply our method to…

We give a new approach to prove the existence of a weak solution of \[dx_t = f(t,x_t)dt + g(t)dB^H_t\] where $B^H_t$ is a fractional Brownian motion with values in a separable Hilbert space for suitable functions $f$ and $g$. Our idea is to…

In this work we consider the energy subcritical 3D wave equation $\partial_t^2 u - \Delta u = \pm |u|^{p-1} u$ and discuss its (weakly) non-radiative solutions, i.e. the solutions defined in an exterior region $\{(x,t): |x|>|t|+R\}$ with…

We study invariant random fields of nonlinear multiplicative stochastic heat equations in the weak disorder regime. Under a natural second-moment condition, we show that positive invariant fields are in one-to-one correspondence with…

A subdiffusion problem in which the diffusion term is related to a stable stochastic process is introduced. Linear models of these systems have been studied in a general way, but non-linear models require a more specific analysis. The model…

In this paper, we investigate Harnack estimates for weak solutions to the following nonlocal equation: $$ \partial_t u = \Delta^{\alpha/2} u + b \cdot \nabla u + f, $$ where $\Delta^{\alpha/2}$ denotes the fractional Laplacian, $b$ is a…

In this paper, we consider the one-dimensional stochastic heat equation driven by a space time white noise. In two different scenarios: {\it (i)} initial condition $u_0=1$ and general nonlinear coefficient $\sigma$ and {\it (ii)}: initial…

In this paper, we investigate the stochastic differential equation on $\mathbb{R}^d,d\geq2$: \begin{align*} \dif X_t&=v(t,X_t)\dif t+\sqrt{2} \dif W_t. \end{align*} For any finite collection of initial probability measures…

In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone…

In this paper, we consider semilinear stochastic fractional heat equation $\frac{\partial}{\partial t}u_{\beta,t}(x)=\triangle^{\alpha/2}u_{\beta,t}(x)+\sigma(u_{\beta,t}(x))\eta_{\beta}$. The Gaussian noise $\eta_{\beta}$ is assumed to be…

We investigate the equation $(u_t + (f(u))_x)_x = f''(u) (u_x)^2/2$ where $f(u)$ is a given smooth function. Typically $f(u)= u^2/2$ or $u^3/3$. This equation models unidirectional and weakly nonlinear waves for the variational wave…

We study existence and regularity of the density for the solution $u(t,x)$ (with fixed $t > 0$ and $x \in D$) of the heat equation in a bounded domain $D \subset \mathbb R^d$ driven by a stochastic inhomogeneous Neumann boundary condition…

In this paper, we establish lower and upper Gaussian bounds for the probability density of the mild solution to the stochastic heat equation with multiplicative noise and in any space dimension. The driving perturbation is a Gaussian noise…

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure…

In this paper, we investigate a semilinear stochastic parabolic equation with a linear rough term $du_{t}=\left[L_{t}u_{t}+f\left(t, u_{t}\right)\right]dt+\left(G_{t}u_{t}+g_{t}\right)d\mathbf{X}_{t}+h\left(t, u_{t}\right)dW_{t}$, where…

Using analysis for 2-admissible functions in weighted Sobolev spaces and stochastic calculus for possibly degenerate symmetric elliptic forms, we construct weak solutions to a wide class of stochastic differential equations starting from an…

In this paper, we obtain the existence and uniqueness of the strong solution to one spatial dimension stochastic wave equation $\frac{\partial^2 u(t,x)}{\partial t^2}=\frac{\partial^2 u(t,x)}{\partial x^2}+\sigma(t,x,u(t,x))\dot{W}(t,x)$…