On a class of exponential changes of measure for stochastic PDEs

Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
{"title":"On a class of exponential changes of measure for stochastic PDEs","authors":"Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart","doi":"arxiv-2409.08057","DOIUrl":null,"url":null,"abstract":"Given a mild solution $X$ to a semilinear stochastic partial differential\nequation (SPDE), we consider an exponential change of measure based on its\ninfinitesimal generator $L$, defined in the topology of bounded pointwise\nconvergence. The changed measure $\\mathbb{P}^h$ depends on the choice of a\nfunction $h$ in the domain of $L$. In our main result, we derive conditions on\n$h$ for which the change of measure is of Girsanov-type. The process $X$ under\n$\\mathbb{P}^h$ is then shown to be a mild solution to another SPDE with an\nextra additive drift-term. We illustrate how different choices of $h$ impact\nthe law of $X$ under $\\mathbb{P}^h$ in selected applications. These include the\nderivation of an infinite-dimensional diffusion bridge as well as the\nintroduction of guided processes for SPDEs, generalizing results known for\nfinite-dimensional diffusion processes to the infinite-dimensional case.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Probability","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08057","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

Abstract

Given a mild solution $X$ to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator $L$, defined in the topology of bounded pointwise convergence. The changed measure $\mathbb{P}^h$ depends on the choice of a function $h$ in the domain of $L$. In our main result, we derive conditions on $h$ for which the change of measure is of Girsanov-type. The process $X$ under $\mathbb{P}^h$ is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of $h$ impact the law of $X$ under $\mathbb{P}^h$ in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.
论随机 PDE 的一类指数量变
给定半线性随机偏微分方程(SPDE)的温和解$X$,我们考虑基于其无限小生成器$L$的指数变化度量,该度量在有界点顺收敛拓扑中定义。变化后的度量 $\mathbb{P}^h$ 取决于在 $L$ 的域中选择一个函数 $h$。在我们的主要结果中,我们推导出了关于$h$ 的条件,在这些条件下,度量的变化属于吉尔萨诺夫类型。然后,我们证明了$X$ 在$mathbb{P}^h$ 下的过程是另一个具有额外加漂移项的 SPDE 的温和解。我们在选定的应用中说明了不同的 $h$ 选择如何影响 $X$ 在 $\mathbb{P}^h$ 下的规律。这些应用包括无穷维扩散桥的衍生,以及引入 SPDE 的引导过程,将已知的无穷维扩散过程的结果推广到无穷维情况。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
求助全文
约1分钟内获得全文 求助全文
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信