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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A random Trotter product formula
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by Thomas G. Kurtz
Proc. Amer. Math. Soc. 35 (1972), 147-154
DOI: https://doi.org/10.1090/S0002-9939-1972-0303347-5

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".
References
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Bibliographic Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 35 (1972), 147-154
  • MSC: Primary 47D05; Secondary 60J35
  • DOI: https://doi.org/10.1090/S0002-9939-1972-0303347-5
  • MathSciNet review: 0303347