Degenerate Kolmogorov equations and ergodicity for the stochastic Allen-Cahn equation with logarithmic potential

Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at t...

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Bibliographic Details
Published inarXiv.org
Main Authors Scarpa, Luca, Zanella, Margherita
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 20.06.2022
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Summary:Well-posedness à la Friedrichs is proved for a class of degenerate Kolmogorov equations associated to stochastic Allen-Cahn equations with logarithmic potential. The thermodynamical consistency of the model requires the potential to be singular and the multiplicative noise coefficient to vanish at the respective potential barriers, making thus the corresponding Kolmogorov equation not uniformly elliptic in space. First, existence and uniqueness of invariant measures and ergodicity are discussed. Then, classical solutions to some regularised Kolmogorov equations are explicitly constructed. Eventually, a sharp analysis of the blow-up rates of the regularised solutions and a passage to the limit with a specific scaling yield existence à la Friedrichs for the original Kolmogorov equation.
ISSN:2331-8422