Pathwise upper semi-continuity of random pullback attractors along the time axis

The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form {At(⋅)}t∈R with each At(⋅) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t↦At(ω) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪s∈IAs(ω), where I⊂R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.