On the stochastic porous medium equation

In this paper, we discuss an initial-boundary value problem and a Cauchy problem for the stochastic porous medium equation. Our basic estimates are based on the known results due to (Math. Sbornik 67 (1965) 609–642, Arch. Rational Mech. Anal. 25 (1967) 64–80). By the procedure developed in (Trans. Amer. Math. Soc. 354 (2001) 1117–1135), we obtain solutions over the given probability space rather than martingale solutions. We will also establish the existence of invariant measures when the space domain is bounded.