A random Trotter product formula

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 ${T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\... ...){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})$ . 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

$n = 0,1,2, \cdots$ the result is just the ``Trotter product formula".