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
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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 ${T_\lambda }(t,w) = {T_{{\xi _0}}}((1/\lambda ){\Delta _0}){T_{{\xi _1}}}((1/\lambda ){\Delta _1}) \cdots T_{\xi _{N(\lambda t)}}((1/\lambda ){\Delta _{\lambda t}})$ . 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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- R. Hersh and M. Pinsky, Random evolutions are asymptotically Gaussian, Comm. Pure Appl. Math. 25 (1972), 33–44. MR 295138, DOI 10.1002/cpa.3160250104
- Thomas G. Kurtz, A general theorem on the convergence of operator semigroups, Trans. Amer. Math. Soc. 148 (1970), 23–32. MR 256210, DOI 10.1090/S0002-9947-1970-0256210-5
- H. F. Trotter, On the product of semi-groups of operators, Proc. Amer. Math. Soc. 10 (1959), 545–551. MR 108732, DOI 10.1090/S0002-9939-1959-0108732-6
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