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Optimal Gaussian density estimates for a class of stochastic equations with additive noise. (English) Zbl 1235.60060
The authors establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. This class includes in particular solutions of SPDEs with additive noise of the form \[ {\mathcal L}u(t,x)=b(u(t,x)) + \sigma \dot W(t,x), \] where \(\mathcal L\) is a second-order space-time linear operator which admits a fundamental solution \(\Gamma\) (in particular \(\mathcal L\) can be of the form \(\partial_t - \partial_{xx}^2\) or \(\partial_{tt}^2 - \partial_{xx}^2\)). The upper and lower Gaussian bounds have the same form and are given in terms of the time variance of the noise.
The proof is based on a suitable application of recent density formulas obtained by I. Nourdin and F. G. Viens [Electron. J. Probab. 14, 2287–2309 (2009; Zbl 1192.60066)].

60H07 Stochastic calculus of variations and the Malliavin calculus
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
Full Text: DOI
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