Instantaneous everywhere-blowup of parabolic SPDEs

IF 1.5 1区数学 Q2 STATISTICS & PROBABILITY

Probability Theory and Related Fields Pub Date : 2024-03-05 DOI:10.1007/s00440-024-01263-7

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引用次数: 0

Abstract

We consider the following stochastic heat equation $$\begin{aligned} \partial _t u(t,x) = \tfrac{1}{2} \partial ^2_x u(t,x) + b(u(t,x)) + \sigma (u(t,x)) {\dot{W}}(t,x), \end{aligned}$$ defined for $(t,x)\in (0,\infty )\times {\mathbb {R}}$ , where ${\dot{W}}$ denotes space-time white noise. The function $\sigma $ is assumed to be positive, bounded, globally Lipschitz, and bounded uniformly away from the origin, and the function b is assumed to be positive, locally Lipschitz and nondecreasing. We prove that the Osgood condition $$\begin{aligned} \int _1^\infty \frac{\textrm{d}y}{b(y)}<\infty \end{aligned}$$ implies that the solution almost surely blows up everywhere and instantaneously, In other words, the Osgood condition ensures that $\textrm{P}\{ u(t,x)=\infty \quad \hbox { for all } t>0 \hbox { and } x\in {\mathbb {R}}\}=1.$ The main ingredients of the proof involve a hitting-time bound for a class of differential inequalities (Remark 3.3), and the study of the spatial growth of stochastic convolutions using techniques from the Malliavin calculus and the Poincaré inequalities that were developed in Chen et al. (Electron J Probab 26:1–37, 2021, J Funct Anal 282(2):109290, 2022).

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抛物线 SPDE 的瞬时无处爆炸

摘要我们考虑以下随机热方程 $$\begin{aligned}\partial _t u(t,x) = \tfrac{1}{2}\partial ^2_x u(t,x) + b(u(t,x))+ sigma (u(t,x)){dot{W}}(t,x), end{aligned}$$ 定义为 $(t,x)\in (0,\infty )\times {mathbb {R}}$其中 ${\dot{W}}$ 表示时空白噪声。函数 $\sigma $被假定为正值、有界、全局 Lipschitz 且远离原点均匀有界，函数 b 被假定为正值、局部 Lipschitz 且不递减。我们证明了奥斯古德条件 $$\begin{aligned}\int _1^\infty \frac{textrm{d}y}{b(y)}<；\換句話說，Osgood 條件確保（textrm{P}{ u(t,x)=\infty \quad \hbox { for all } t>0 \hbox { and } xin {\mathbb {R}}\}=1.\) 证明的主要内容涉及一类微分不等式的命中时间约束（备注 3.3），以及利用马利亚文微积分和陈等人的 Poincaré 不等式（Electron J Probab 26:1-37, 2021, J Funct Anal 282(2):109290, 2022）所发展的技术研究随机卷积的空间增长。

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来源期刊

Probability Theory and Related Fields 数学-统计学与概率论

CiteScore

3.70

自引率

5.00%

发文量

审稿时长

6-12 weeks

期刊介绍： Probability Theory and Related Fields publishes research papers in modern probability theory and its various fields of application. Thus, subjects of interest include: mathematical statistical physics, mathematical statistics, mathematical biology, theoretical computer science, and applications of probability theory to other areas of mathematics such as combinatorics, analysis, ergodic theory and geometry. Survey papers on emerging areas of importance may be considered for publication. The main languages of publication are English, French and German.