On the 𝐿_{𝑝}-boundedness of the stochastic singular integral operators and its application to 𝐿_{𝑝}-regularity theory of stochastic partial differential equations,Transactions of the American Mathematical Society

当前位置： X-MOL 学术 › Trans. Am. Math. Soc. › 论文详情

Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)

On the 𝐿_{𝑝}-boundedness of the stochastic singular integral operators and its application to 𝐿_{𝑝}-regularity theory of stochastic partial differential equations
Transactions of the American Mathematical Society ( IF 1.2 ) Pub Date : 2020-05-26 , DOI: 10.1090/tran/8089
Ildoo Kim , Kyeong-Hun Kim

Abstract:In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let

be a Wiener process in a probability space $\Omega$ and let $K(\omega ,r,t,x,y)$ be a random kernel which is allowed to be stochastically singular in a domain $\mathcal {O} \subset \mathbf {R}^d$ in the sense that

$\displaystyle \mathbb{E} \left \vert\int _0^{t} \int _{\vert x-y\vert<\varepsil... ...ght \vert^p = \infty \quad \forall \, t, p,\varepsilon >0,\, x\in \mathcal {O}.$

We prove that the stochastic integral operator of the type

$\displaystyle \mathbb{T} g(t,x) \colonequals \int _0^{t} \int _{\mathcal {O}} K(\omega ,s,t,y,x) g(s,y)dy dW_s$

(1)

is bounded on $\mathbb{L}_p=L_p \left (\Omega \times (0,\infty ); L_{p}(\mathcal {O}) \right )$ for all $p \in [2,\infty )$ if it is bounded on $\mathbb{L}_2$ and the following (which we call stochastic Hörmander condition) holds: there exists a quasi-metric $\rho$ on $(0,\infty )\times \mathcal {O}$ and a positive constant

such that for $X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty ) \times \mathcal {O}$ ,

$\displaystyle \sup _{\omega \in \Omega ,X,Y}\int _{0}^\infty \left [ \int _{\rh... ...C_0 \rho (X,Y)} \vert K(r,t, z,x) - K(r,s, z,y)\vert ~dz\right ]^2 dr <\infty .$

Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp

-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.

中文翻译：

随机奇异积分算子的𝐿_{𝑝}有界性及其在随机偏微分方程的𝐿_{𝑝}正则性理论中的应用

摘要：在本文中，我们介绍了Hörmander条件和Calderón-Zygmund定理的随机对应项。让

是一个概率空间维纳过程 $\ Omega$ ，让是被允许是随机单一域中的随机内核在这个意义上， $K（\ omega，r，t，x，y）$ $\ mathcal {O} \ subset \ mathbf {R} ^ d$

$\ displaystyle \ mathbb {E} \ left \ vert \ int _0 ^ {t} \ int _ {\ vert xy \ vert <\ varepsil ... ... ght \ vert ^ p = \ infty \ quad \ forall \ ，\ t，p，\ varepsilon> 0，\，x \ in \ mathcal {O}。$

我们证明了这种类型的随机积分算子

$\ displaystyle \ mathbb {T} g（t，x）\ colonequals \ int _0 ^ {t} \ int _ {\ mathcal {O}} K（\ omega，s，t，y，x）g（s， y）dy dW_s$

（1）

上界所有，如果它为界，与以下（我们称之为随机Hörmander条件）成立：存在一个准度上和正常数，使得对于， $\ mathbb {L} _p = L_p \ left（\ Omega \ times（0，\ infty）; L_ {p}（\ mathcal {O}）\ right）$ $p \ in [2，\ infty）$ $\ mathbb {L} _2$ $\ rho$ $（0，\ infty）\ times \ mathcal {O}$

$X =（t，x），Y =（s，y），Z =（r，z）\ in（0，\ infty）\ times \ mathcal {O}$

$\ displaystyle \ sup _ {\ omega \ in \ Omega，X，Y} \ int _ {0} ^ \ infty \ left [\ int _ {\ rh ... ... C_0 \ rho（X，Y） } \ vert K（r，t，z，x）-K（r，s，z，y）\ vert〜dz \ right] ^ 2 dr <\ infty。$

当证明随机偏微分方程（SPDE）的解的最大正则性时，这种随机奇异积分自然会出现。作为应用程序，我们获得了大量

SPDE的尖锐正则性结果，其中包括具有时间可测量伪微分算子的SPDE和在非平滑角域上定义的SPDE。

更新日期：2020-05-26

点击分享查看原文

点击收藏

公开下载

阅读更多本刊最新论文本刊介绍/投稿指南11