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Quasi-invariance for families of infinite-dimensional SDEs with degenerate noise

by Hyoungji https://orcid.org/0000-0001-7161-4956 Kim

Institution:	Iowa State University
Department:	Mathematics
Degree:	PhD
Year:	2022
Keywords:	Mathematics; gradient estimates; hypoellipticity; infinite-dimensional process; Markov process; quasi-invariance; stochastic differential equation
Posted:	3/25/2025
Record ID:	2292656
Full text PDF:	https://dr.lib.iastate.edu/handle/20.500.12876/kv7kj0Pv

Abstract

We introduce quantitative functional inequalities for specific classes of hypoelliptic stochastic differential equations (SDEs). In particular, these inequalities include the reverse logarithmic Sobolev inequality and Wang-type Harnack inequality for a large class of linear SDEs with degenerate noise. The inequalities are obtained from gradient bounds of the semigroup of the process and by making use of the generalized version of the carre du champ operator. From these results, we obtain quasi-invariance of the same form of SDEs that take values in an infinite-dimensional Hilbert space, deducing a form of hypoellipticity in infinite dimensions.