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 ut 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.
Robert C. Dalang. C. Mueller. L. Zambotti. "Hitting properties of parabolic s.p.d.e.’s with reflection." Ann. Probab. 34 (4) 1423 - 1450, July 2006. https://doi.org/10.1214/009117905000000792