Resumen de Integration by parts on \boldsδ-Bessel bridges, \boldsδ>3, and related SPDEs

Lorenzo Zambotti

• We study a white-noise driven semilinear partial differential equation on the spatial interval [0,1] with Dirichlet boundary condition and with a singular drift of the form cu−3, c>0. We prove existence and uniqueness of a non-negative continuous adapted solution u on [0,∞)×[0,1] for every nonnegative continuous initial datum x, satisfying x(0)=x(1)=0. We prove that the law πδ of the Bessel bridge on [0,1] of dimension δ>3 is the unique invariant probability measure of the process x↦u, with c=(δ−1)(δ−3)/8 and, if δ∈N, that u is the radial part in the sense of Dirichlet forms of the Rδ-valued solution of a linear stochastic heat equation. An explicit integration by parts formula w.r.t. πδ is given for all δ>3.