Speaker
Description
We study a stochastic reaction–diffusion equation on the whole real line, driven by space–time white noise, with a possibly nonlocal diffusion operator and a superquadratic potential. Unlike the case of bounded intervals, on the whole line the invariant measures cannot be described by an explicit density with respect to a Gaussian measure. We formulate the equation in suitable weighted spaces and show that invariant and infinitesimally invariant measures coincide, and that reversible measures coincide with the solutions of an integration-by-parts formula. Finally, we prove that the reversible measures are precisely the Gibbs measures with finite moments defined via the Dobrushin–Lanford–Ruelle equations. In particular, we connect the Da Prato–Zabczyk approach to invariant measures of SPDEs with the statistical mechanics approach of Osada and Spohn, within a framework covering both uniqueness and non-uniqueness regimes. Joint work with F. C. De Vecchi, C. Orrieri and C. Villanueva Mariz.