Absolute continuity for SPDEs with irregular fundamental solution. (English) Zbl 1321.60138

Sanz-Solé, Marta; Süss, André

Electron. Commun. Probab. 20, Paper No. 14, 11 p. (2015).

Summary: For a class of stochastic partial differential equations studied by D. Conus and R. C. Dalang [Electron. J. Probab. 13, 629–670 (2008; Zbl 1187.60049)], we prove the existence of the density of the probability law of the solution at a given point \((t,x)\), and that the density belongs to some Besov space. The proof relies on a method developed by A. Debussche and M. Romito [Probab. Theory Relat. Fields 158, No.3–4, 575–596 (2014; Zbl 1452.76193)]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension \(d\geq 1\), and also to the heat equation. This provides an extension of earlier results.

MSC:

60H15	Stochastic partial differential equations (aspects of stochastic analysis)
60H07	Stochastic calculus of variations and the Malliavin calculus
60H20	Stochastic integral equations
60H05	Stochastic integrals
35R60	PDEs with randomness, stochastic partial differential equations

Keywords:

stochastic partial differential equations; irregular fundamental solution; densities; absolute continuity; stochastic wave equation

Citations:

Zbl 1187.60049; Zbl 1452.76193