BV functions in Hilbert spaces

We study the basic theory of BV functions in a Hilbert space X endowed with a (not necessarily Gaussian) probability measure ν . We present necessary and sufficient conditions in order that a function u ∈ L p ( X , ν ) is of bounded variation. We also discuss the De Giorgi approach to BV functions t...

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Bibliographic Details
Published in:Mathematische annalen 2021-12, Vol.381 (3-4), p.1653-1722
Main Authors: Da Prato, Giuseppe, Lunardi, Alessandra
Format: Article
Language:English
Subjects:
Burgers equation
Hilbert space
Mathematics
Mathematics and Statistics
Normal distribution
Reaction-diffusion equations
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Summary:We study the basic theory of BV functions in a Hilbert space X endowed with a (not necessarily Gaussian) probability measure ν . We present necessary and sufficient conditions in order that a function u ∈ L p ( X , ν ) is of bounded variation. We also discuss the De Giorgi approach to BV functions through the behavior as t → 0 of ∫ X ‖ ∇ T ( t ) u ‖ d ν , for a smoothing semigroup T ( t ). Particular attention is devoted to the case where u is the indicator function of a sublevel set { x : g ( x ) < r } of a real Borel function g . We give several examples, for different measures ν such as weighted Gaussian measures, infinite products of non Gaussian measures, and invariant measures of some stochastic PDEs such as reaction-diffusion equations and Burgers equation.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-020-02037-x