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.Comment: Published at http://dx.doi.org/10.1214/009117905000000792 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org