A Central Limit Problem in Random Evolutions

Abstract

Let $\{T_n\}_{n \geq 1}$ be a sequence of independent and identically distributed strongly continuous semigroups on a separable Banach space. The corresponding generators $\{A_n\}_{n \geq 1}$ satisfy $E\lbrack A_n\rbrack = 0$. Conditions are given to guarantee that the weak limit $Y(t) = \text{limit}_{n \rightarrow \infty} \prod^{\lbrack n^2t\rbrack}_{i = 1} T_i(1/n) Y_n(0)$ exists, and is characterized as the unique solution of a martingale problem. Transport phenomena, random classical mechanics, and families of bounded operators are the featured examples.