Long-time behaviour of nonautonomous SPDEʹs

It is proved that under suitable conditions the probability laws of two arbitrary solutions of the infinite dimensional stochastic equationdXt=AXt dt+f(t,Xt) dt+Q1/2 dWtconverge to each other, as time goes to infinity, in the strong (variational) topology. To this end, some lower estimates on the transition density of the solution, with respect to a certain Gaussian measure, are obtained. In addition, an explicit formula for the density is given, in the case where Q−1/2f is bounded.