A random Trotter product formula

Abstract:

Let $X(t)$ be a pure jump process with state space S and let ${\xi _0},{\xi _1},{\xi _2}, \cdots$ be the succession of states visited by $X(t),{\Delta _0}{\Delta _1} \cdots$ the sojourn times in each state, $N(t)$ the number of transitions before t and ${\Delta _t} = t - \sum \nolimits _{k = 0}^{N(t) - 1} {{\Delta _k}}$. For each $x \in S$ let ${T_x}(t)$ be an operator semigroup on a Banach space L. Define ${T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\lambda ){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})$ . Conditions are given under which ${T_\lambda }(t,w)$ will converge almost surely (or in probability) to a semigroup of operators as $\lambda \to \infty$. With $S = \{ 1,2\}$ and \[ \begin {array}{*{20}{c}} \hfill {X(t) = 1,\quad } \\ \hfill { = 2,\quad } \\ \end {array} \begin {array}{*{20}{c}} {2n \leqq t < 2n + 1,} \hfill \\ {2n + 1 \leqq t < 2n + 2,} \hfill \\ \end {array} \] $n = 0,1,2, \cdots$ the result is just the “Trotter product formula".