On minimal parabolic functions and time-homogeneous parabolic h-transforms

Time-Homogeneous Parabolic Wick-Anderson Model in One Space Dimension: Regularity of Solution

Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.

Hölder continuity for the Parabolic Anderson Model with space-time homogeneous Gaussian noise

In this article, we consider the Parabolic Anderson Model with constant initial condition, driven by a space-time homogeneous Gaussian noise, with general covariance function in time and spatial spectral measure satisfying Dalang's condition. First, we prove that the solution (in the Skorohod sense) exists and is continuous in $L^p(Ω)$. Then, we show that the solution has a modification whose sample paths are Hölder continuous in space and time, with optimal exponents, and under the minimal condition on the spatial spectral measure of the noise (which is the same as the condition encountered i…

Parabolic Anderson Model with space-time homogeneous Gaussian noise and rough initial condition

In this article, we study the Parabolic Anderson Model driven by a space-time homogeneous Gaussian noise on $\mathbb{R}_{+} \times \mathbb{R}^d$, whose covariance kernels in space and time are locally integrable non-negative functions, which are non-negative definite (in the sense of distributions). We assume that the initial condition is given by a signed Borel measure on $\mathbb{R}^d$, and the spectral measure of the noise satisfies Dalang's (1999) condition. Under these conditions, we prove that this equation has a unique solution, and we investigate the magnitude of the $p$-th moments of …

Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour

We consider the solution to the parabolic Anderson model with homogeneous initial condition in large time-dependent boxes. We derive stable limit theorems, ranging over all possible scaling parameters, for the rescaled sum over the solution depending on the growth rate of the boxes. Furthermore, we give sufficient conditions for a strong law of large numbers.

齐次李群上的加权Besov空间及其在抛物Anderson模型中的应用

Weighted Besov Spaces on Homogeneous Lie Groups and Applications to Parabolic Anderson Models

本文在齐次李群上建立加权非齐次Besov空间的统一理论，将其应用于含Rockland算子与奇异初始数据的抛物Anderson型方程，分别在时空噪声和纯空间噪声下得到相关结果，依赖温和缝合引理完成推导。

Time correlations for the parabolic Anderson model

We derive exact asymptotics of time correlation functions for the parabolic Anderson model with homogeneous initial condition and time-independent tails that decay more slowly than those of a double exponential distribution and have a finite cumulant generating function. We use these results to give precise asymptotics for statistical moments of positive order. Furthermore, we show what the potential peaks that contribute to the intermittency picture look like and how they are distributed in space. We also investigate for how long intermittency peaks remain relevant in terms of ageing properti…