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Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs. (English) Zbl 1499.60234

Summary: We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in [T. Funaki et al., Ann. Inst. Henri Poincaré, Probab. Stat. 57, No. 3, 1702–1735 (2021; Zbl 1484.60066)], 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 \(t\rightarrow \infty\). 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 [loc. cit.] except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.

MSC:

60H17 Singular stochastic partial differential equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35K67 Singular parabolic equations

Citations:

Zbl 1484.60066
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References:

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