2018/02/18 by Dimitra C. Antonopoulou, Antonopoulou, D. C., D. Farazakis +3 · 1 citation
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Solidification and crystal growth phenomena #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1802.06389
openalex publication_date 2018/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The stochastic partial differential equation analyzed in this work, is motivated by a simplified mesoscopic physical model for phase separation. It describes pattern formation due to adsorption and desorption mechanisms involved in surface processes, in the presence of a stochastic driving force. This equation is a combination of Cahn-Hilliard and Allen-Cahn type operators with a multiplicative, white, space-time noise of unbounded diffusion. We apply Malliavin calculus, in order to investigate the existence of a density for the stochastic solution u. In dimension one, according to the regularity result in \citeAKM, u admits continuous paths a.s. Using this property, and inspired by a method proposed in \citeCW1, we construct a modified approximating sequence for u, which properly treats the new second order Allen-Cahn operator. Under a localization argument, we prove that the Malliavin derivative of u exists locally, and that the law of u is absolutely continuous, establishing thus that a density exists.