Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

Tadahisa Funaki*, Bin Xie

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021), which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as t→ ∞. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in Funaki et al. (Ann Inst Henri Poincaré Probab Stat 57:1702-1735, 2021) except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.

Original languageEnglish
JournalStochastics and Partial Differential Equations: Analysis and Computations
DOIs
Publication statusAccepted/In press - 2022

Keywords

  • Energy inequality
  • Global solvability
  • Paracontrolled calculus
  • Quasilinear SPDE
  • Singular SPDE
  • Stationary solution

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics

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