A probabilistic Harnack inequality and strict positivity of stochastic partial differential equations (Q1656536)

A priori probabilistic Harnack inequality for the non-negative solution of a semi-linear SPDE of the following form \[ \partial_t u=\mathrm{div}(\mathbb A\nabla u)+f(t, x, u; \omega)+g_i(t, x, u; \omega)\dot{w}^i_t \] is established under general conditions. Also, the almost surely strictly positive of the solutions for this model is shown when the initial condition is non-negative and not identically vanishing. The article will be very useful for researches in stochastic analysis for stochastic partial differential equations.