A random product of markovian semi-groups of operators

Abstract

Let Y(t) be a continuous time pure jump process with state space S = {1,2,...,n} and let ξ₀,ξ₁,..., be the succession of states visited by Y(t), Δ₀,Δ₁,... the sojourn times in each state, N(t) the number of transitions before t and $\Delta _t = t - \sum\limits_{k = 0}^{N(t) - 1} {\Delta _k }$. For each k ε S, let T_k(t) be an operator semigroup on a Banach space L with infinitesimal generator a_k. Define T_λ (t,ω)=$T_{\xi _0 } (\tfrac{1}{\lambda }\Delta _0 )T_{\xi _1 } (\tfrac{1}{\lambda }\Delta _1 ) \cdot ... \cdot T_{\xi _{{\rm N}(\lambda t)} } (\tfrac{1}{\lambda }\Delta _{\lambda t} )$. It is known (Kurtz) that if

exist for all k = 1,2,...,n and $\sum\limits_{i = 1}^n {\mu _i = 1}$, then under appropriate conditions the closure of a = $\sum\limits_{i = 1}^n {\mu _i a_i }$is the infinitesimal operator for a strongly continuous semigroup T(t) defined on L and T_λ(t,ω) converges almost surely to T(t) as λ → ∞. The existence and identification of this limit is of interest even when the closure of a is not a generator. A probabilistic version of this problem is given in the case of Markovian transition semi-group when the corresponding processes have identical hitting distributions. Sufficient conditions for the existence of limit are given. With S = {1,2} and $\left\{ \begin{gathered}1,2n \leqslant t < 2n + 1 \hfill \\2,2n + 1 \leqslant t < 2n + 2 \hfill \\\end{gathered} \right.$, n = 0,1,2,..., the result is obtained by Loren Pitt.