全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...
Mathematics  2014 

Large deviations for white-noise driven, nonlinear stochastic PDEs in two and three dimensions

Full-Text   Cite this paper   Add to My Lib

Abstract:

We study the stochastic Allen-Cahn equation driven by a noise term with intensity $\sqrt{\varepsilon}$ and correlation length $\delta$ in two and three spatial dimensions. We study diagonal limits $\delta, \varepsilon \to 0$ and describe fully the large deviation behaviour depending on the relationship between $\delta$ and $\varepsilon$. The recently developed theory of regularity structures allows to fully analyse the behaviour of solutions for vanishing correlation length $\delta$ and fixed noise intensity $\varepsilon$. One key fact is that in order to get non-trivial limits as $\delta \to 0$, it is necessary to introduce diverging counterterms. The theory of regularity structures allows to rigorously analyse this renormalisation procedure for a number of interesting equations. Our main result is a large deviation principle for these renormalised solutions. One interesting feature of this result is that the diverging renormalisation constants disappear at the level of the large deviations rate function. We apply this result to derive a sharp condition on $\delta, \varepsilon$ that guarantees a large deviation principle for diagonal schemes $\varepsilon, \delta \to 0$ for the equation without renormalisation.

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133