Strong convergence rate in averaging principle for stochastic hyperbolic-parabolic equations with two time-scales

双曲空间中的抛物型 Anderson 模型。第一部分：退火渐近性

Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics

本文研究了双曲空间中具有高斯势的抛物型 Anderson 模型的二阶矩渐近性，发现其增长、波动及间歇性均与欧氏情形相同，并指出波动指数由欧氏拉普拉斯算子诱导的变分问题决定，这源于重整化后的曲率膨胀效应。

高斯色噪声驱动的抛物/双曲 Anderson 模型的几乎必然中心极限定理

Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises

本文针对时间着色高斯噪声驱动的抛物/双曲 Anderson 模型，结合二阶高斯 Poincaré 不等式与特征函数方法，建立了几乎必然中心极限定理，完善了随机偏微分方程的定量中心极限理论。

Well-posedness and regularity for quasilinear degenerate parabolic-hyperbolic SPDE

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions.

带Levy噪声的退化抛物-双曲方程的重整化随机熵解

Renormalized stochastic entropy solution for degenerate parabolic-hyperbolic equations with Levy noise

本文针对无界区域上由乘性Levy噪声扰动且带一般L1-data的退化抛物-双曲PDE，利用消失粘性技术证明重整化熵解的存在性，并采用Kružkov变量倍增技术证明解的唯一性，建立了适定性理论。

Stochastic non-isotropic degenerate parabolic-hyperbolic equations

We introduce the notion of pathwise entropy solutions for a class of degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity and fluxes with rough time dependence and prove their well-posedness. In the case of Brownian noise and periodic boundary conditions, we prove that the pathwise entropy solutions converge to their spatial average and provide an estimate on the rate of convergence. The third main result of the paper is a new regularization result in the spirit of averaging lemmata. This work extends both the framework of pathwise entropy solutions for stochastic scalar c…

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and d…

The Martin boundary of relatively hyperbolic groups with virtually abelian parabolic subgroups

Given a probability measure on a finitely generated group, its Martin boundary is a way to compactify the group using the Green's function of the corresponding random walk. We give a complete topological characterization of the Martin boundary of finitely supported random walks on relatively hyperbolic groups with virtually abelian parabolic subgroups. In particular, in the case of nonuniform lattices in the real hyperbolic space H n , we show that the Martin boundary coincides with the CAT (0) boundary of the truncated space, and thus when n = 3, is homeomorphic to the Sierpinski carpet.

Nonlinear Anisotropic Degenerate Parabolic-Hyperbolic Equations with Stochastic Forcing

We are concerned with nonlinear anisotropic degenerate parabolic-hyperbolic equations with stochastic forcing, which are heterogeneous (i.e., not space-translational invariant). A unified framework is established for the continuous dependence estimates, fractional BV regularity estimates, and well-posedness for stochastic entropy solutions of the nonlinear stochastic degenerate parabolic-hyperbolic equation. In particular, we establish the well-posedness of the nonlinear stochastic equation in $L^p \cap N^{κ,1}$ for $p\in (1,\infty)$ and the $κ$--Nikolskii space $N^{κ,1}$ with $κ>0$, and the $…

Central limit theorem for full discretization of parabolic SPDE

In order to characterize the fluctuation between the ergodic limit and the time-averaging estimator of a full discretization in a quantitative way, we establish a central limit theorem for the full discretization of the parabolic stochastic partial differential equation. The theorem shows that the normalized time-averaging estimator converges to a normal distribution with the variance being the same as that of the continuous case, where the scale used for the normalization corresponds to the temporal strong convergence order of the considered full discretization. A key ingredient in the proof …

双曲空间中的抛物型安德森模型。第二部分：淬火渐进行为

Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics

该研究在双曲空间中分析了抛物型安德森模型的淬火渐进行为，揭示了指数增长的幂律和系数的确定方法。

On the parabolic and hyperbolic Liouville equations

We study the two-dimensional stochastic nonlinear heat equation (SNLH) and stochastic damped nonlinear wave equation (SdNLW) with an exponential nonlinearity $λβe^{βu }$, forced by an additive space-time white noise. We prove local and global well-posedness of these equations, depending on the sign of $λ$ and the size of $β^2 > 0$, and invariance of the associated Gibbs measures. See the abstract of the paper for a more precise abstract. (Due to the limit on the number of characters for an abstract set by arXiv, the full abstract can not be displayed here.)