Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
{"title":"论随机 PDE 的一类指数量变","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":"{\"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}","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}
On a class of exponential changes of measure for stochastic PDEs
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.