On divergence form SPDEs with VMO coefficients

Strong solutions to SPDEs with monotone drift in divergence form

We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, an…

SPDEs驱动的分歧自由传输噪声下的遍历性

Ergodicity for SPDEs driven by divergence-free transport noise

本文研究了由分歧自由传输噪声驱动的SPDEs的遍历性，证明强混合噪声可确保唯一不变测度，即使确定性部分存在多个稳态。

An $L^p$-theory of non-divergence form SPDEs driven by Lévy processes

In this paper we present an $L^p$-theory for the stochastic partial differential equations (SPDEs in abbreciation) driven by Lé{}vy processes. Existence and uniqueness of solutions in Sobolev spaces are obtained. The coefficients of SPDEs under consideration are random functions depending on time and space variables.

On the well-posedness of SPDEs with singular drift in divergence form

We prove existence and uniqueness of strong solutions for a class of second-order stochastic PDEs with multiplicative Wiener noise and drift of the form $\operatorname{div} γ(\nabla \cdot)$, where $γ$ is a maximal monotone graph in $\mathbb{R}^n \times \mathbb{R}^n$ obtained as the subdifferential of a convex function satisfying very mild assumptions on its behavior at infinity. The well-posedness result complements the corresponding one in our recent work arXiv:1612.08260 where, under the additional assumption that $γ$ is single-valued, a solution with better integrability and regularity prop…

Estimates in $L_{p}$ for solutions of SPDEs with coefficients in Morrey classes

For solutions of a certain class of SPDEs in divergence form we present some estimates of their $L_{p}$-norms and the $L_{p}$-norms of their first-order derivatives. The main novelty is that the low-order coefficients are supposed to belong to certain Morrey classes instead of $L_{p}$-spaces. Our results are new even if there are no stochastic terms in the equation.

奇异SPDE的均匀化

Homogenisation of singular SPDEs

本研究提出一种结合经典均匀化结果与正则性结构的方法，研究奇异SPDE的均匀化，建立了二维g-PAM与Φ⁴₃方程振荡推广的周期均匀化结果，明确了发散项结构与所需的重整化常数。

Spatial quadratic variations for the solution to a stochastic partial differential equation with elliptic divergence form operator

We introduce a stochastic partial differential equation (SPDE) with elliptic operator in divergence form, with measurable and bounded coefficients and driven by space-time white noise. Such SPDEs could be used in mathematical modelling of diffusion phenomena in medium consisting of different kinds of materials and undergoing stochastic perturbations. We characterize the solution and, using the Stein--Malliavin calculus, we prove that the sequence of its recentered and renormalized spatial quadratic variations satisfies an almost sure central limit theorem. Particular focus is given to the inte…

A Sobolev Space theory for stochastic partial differential equations with time-fractional derivatives

In this article we present an $L_p$-theory ($p\geq 2$) for the time-fractional quasi-linear stochastic partial differential equations (SPDEs) of type $$ \partial^α_tu=L(ω,t,x)u+f(u)+\partial^β_t \sum_{k=1}^{\infty}\int^t_0 ( Λ^k(ω,t,x)u+g^k(u))dw^k_t, $$ where $α\in (0,2)$, $β<α+\frac{1}{2}$, and $\partial^α_t$ and $\partial^β_t$ denote the Caputo derivative of order $α$ and $β$ respectively. The processes $w^k_t$, $k\in \mathbb{N}=\{1,2,\cdots\}$, are independent one-dimensional Wiener processes defined on a probability space $Ω$, $L$ is a second order operator of either divergence or non-div…

Stochastic maximal regularity for rough time-dependent problems

We unify and extend the semigroup and the PDE approaches to stochastic maximal regularity of time-dependent semilinear parabolic problems with noise given by a cylindrical Brownian motion. We treat random coefficients that are only progressively measurable in the time variable. For $2m$-th order systems with $VMO$ regularity in space, we obtain $L^{p}(L^{q})$ estimates for all $p>2$ and $q\geq 2$, leading to optimal space-time regularity results. For second order systems with continuous coefficients in space, we also include a first order linear term, under a stochastic parabolicity condition,…

时间和空间上具有有界可测系数的随机与确定性抛物方程：适定性与极大正则性

Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: well-posedness and maximal regularity

本文针对散度形式线性随机抛物方程，在系数仅需有界可测的广泛条件下，通过引入路径弱解和新的调和分析工具，建立了适定性与极大正则性，并将Lions定理从T²推广至T^p。