Long Time Behavior of Stochastic Thin Film Equation

Oleksiy Kapustyan; Olha Martynyuk; Oleksandr Misiats; Oleksandr Stanzhytskyi

Long Time Behavior of Stochastic Thin Film Equation

Analysis of PDEs 2026-04-14 v1 Probability

Authors: Oleksiy Kapustyan , Olha Martynyuk , Oleksandr Misiats , Oleksandr Stanzhytskyi

Abstract

We consider the stochastic thin-film equation with linear deterministic and stochastic It\^o perturbations. The existence of nonnegative weak martingale solutions on the semi-axis is established, and their asymptotic behavior as $t \to \infty$ is investigated. It is shown that in square mean the $L^\infty$ norm of the solution converges to the spatial mean value of the initial condition, multiplied by a random factor similar to a geometric Wiener process.

Keywords

stochastic thin film equation martingale solution long-time behavior L-infinity norm geometric Wiener process

Cite

@article{arxiv.2604.10010,
  title  = {Long Time Behavior of Stochastic Thin Film Equation},
  author = {Oleksiy Kapustyan and Olha Martynyuk and Oleksandr Misiats and Oleksandr Stanzhytskyi},
  journal= {arXiv preprint arXiv:2604.10010},
  year   = {2026}
}

Related papers

View all related →