2016/09/27 by Pavlos Tsatsoulis, Tsatsoulis, Pavlos, Hendrik Weber +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · #37A25 #60H15 #81T08 #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth #Probability (math.PR) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.1609.08447
openalex publication_date 2016/09/27 · openalex created_date 2022/08/15 · openalex updated_date 2026/07/28
We study the long time behavior of the stochastic quantization equation.\nExtending recent results by Mourrat and Weber we first establish a strong\nnon-linear dissipative bound that gives control of moments of solutions at all\npositive times independent of the initial datum. We then establish that\nsolutions give rise to a Markov process whose transition semigroup satisfies\nthe strong Feller property. Following arguments by Chouk and Friz we also prove\na support theorem for the laws of the solutions. Finally all of these results\nare combined to show that the transition semigroup satisfies the Doeblin\ncriterion which implies exponential convergence to equilibrium.\n Along the way we give a simple direct proof of the Markov property of\nsolutions and an independent argument for the existence of an invariant measure\nusing the Krylov-Bogoliubov existence theorem. Our method makes no use of the\nreversibility of the dynamics or the explicit knowledge of the invariant\nmeasure and it is therefore in principle applicable to situations where these\nare not available, e.g. the vector-valued case.\n