Hitting properties of parabolic s.p.d.e.’s with reflection

Abstract

We study the hitting properties of the solutions u of a class of parabolic stochastic partial differential equations with singular drifts that prevent u from becoming negative. The drifts can be a reflecting term or a nonlinearity cu⁻³, with c>0. We prove that almost surely, for all time t>0, the solution u_t hits the level 0 only at a finite number of space points, which depends explicitly on c. In particular, this number of hits never exceeds 4 and if c>15/8, then level 0 is not hit.