We investigate the relations between Pesin–Pitskel topological pressure on an arbitrary subset and measure-theoretic pressure of Borel probability measures for finitely generated semigroup actions.Let(X,G)be a system,where X is a compact metric space and
Let u={u(t,x),t∈[0,T],x∈R}be a solution to a stochastic heat equation driven by a space-time white noise.We study that the realized power variation of the process u with respect to the time,properly normalized,has Gaussian asymptotic distributions.In part