Abstract
AbstractWell-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.
Funder
Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni
Ministero dell’Universitá e della Ricerca
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Modeling and Simulation,Statistics and Probability
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