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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A random Trotter product formula

Author: Thomas G. Kurtz
Journal: Proc. Amer. Math. Soc. 35 (1972), 147-154
MSC: Primary 47D05; Secondary 60J35
MathSciNet review: 0303347
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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/\... ...){\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{displaymath}\begin{array}{*{20}{c}} \hfill {X(t) = 1,\quad } \\ \hfill { ... ...1,} \hfill \\ {2n + 1 \leqq t < 2n + 2,} \hfill \\ \end{array} \end{displaymath}

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

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Keywords: Operator semigroup, random evolution, Markov process, Trotter product
Article copyright: © Copyright 1972 American Mathematical Society

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