相关论文: Existence and space-time regularity for stochastic…
We consider a general class of $L^2$-valued stochastic processes that arise primarily as solutions of parabolic SPDEs on p.c.f. fractals. Using a Kolmogorov-type continuity theorem, conditions are found under which these processes admit…
We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to…
A p.c.f. fractal with a regular harmonic structure admits an associated Dirichlet form, which is itself associated with a Laplacian. This Laplacian enables us to give an analogue of the damped stochastic wave equation on the fractal. We…
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique…
A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is…
We study the martingale formulation of the two-dimensional stochastic heat equation (SHE) at criticality. The main theorem proves an exact recursive-type equation that expresses the covariation measures of the SHE in terms of the solutions…
We investigate the optimal H\"older continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index…
The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified Stochastic Heat Equation (SHE), specifically at the critical dimension of two and near the critical temperature.…
Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature.…
It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have random-field solutions only in spatial dimension one. Here we show that in many cases, where the…
We study the large-scale dynamics of the solution to a nonlinear stochastic heat equation (SHE) in dimensions $d \geq 3$ with long-range dependence. This equation is driven by multiplicative Gaussian noise, which is white in time and…
We study the Allen-Cahn equation with a cubic-quintic nonlinear term and a stochastic $Q$-trace-class stochastic forcing in two spatial dimensions. This stochastic partial differential equation (SPDE) is used as a test case to understand,…
This paper constructs a solvability theory for a system of stochastic partial differential equations. On account of the Kolmogorov continuity theorem, solutions are looked for in certain H\"older-type classes in which a random field is…
In this paper, we study the stochastic heat equation (SHE) on $\mathbb{R}^d$ subject to a centered Gaussian noise that is white in time and colored in space. We establish the existence and uniqueness of the random field solution in the…
We consider a generalized model of random walk in dynamical random environment, and we show that the multiplicative-noise stochastic heat equation (SHE) describes the fluctuations of the quenched density at a certain precise location in the…
Higher order fluctuation expansions for stochastic heat equations (SHE) with nonlinear, non-conservative and conservative noise are obtained. These Edgeworth-type expansions describe the asymptotic behavior of solutions in suitable joint…
In this contribution, we provide convergence rates for a finite volume scheme of the stochastic heat equation with multiplicative Lipschitz noise and homogeneous Neumann boundary conditions (SHE). More precisely, we give an error estimate…
We consider the mollified versions of the Kardar-Parisi-Zhang (KPZ) equation and the stochastic heat equation (SHE) in high dimensions $d\geq 3$ and analyze their probability distributions as the mollification is removed. Up to the…
We study the solutions of the stochastic heat equation driven by spatially inhomogeneous multiplicative white noise based on a fractal measure. We prove pathwise uniqueness for solutions of this equation when the noise coefficient is…
We consider the stochastic heat equation of the following form \frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) + \sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the generator of a L\'evy process and…