Summary
Let A and B denote the generators of two contraction semi-groups of operators (P t) and (Q t) acting on some Banach space. If the operator A + B has a closure generating a third semi-group (R t), then it is known (Trotter) that \(R^t = \mathop {\lim }\limits_{h \to 0} (P^h Q^h )^{[t/h]}\). The existence and identification of this limit is of interest even when the closure of A + B is not a generator. A probabilistic version of this problem is given here in the case of Markovian transition semi-groups when the corresponding processes have identical hitting distributions. Sufficient conditions for the existence of (R t) are given, and in special cases its generator is identified.
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This paper is part of a Ph. D. thesis written at Princeton University. The research was done with the partial support of the Office of Army Research. The author also wishes to thank his thesis supervisor Professor W. Feller for his advice and interest in this work.
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Pitt, L. Products of Markovian semi-groups of operators. Z. Wahrscheinlichkeitstheorie verw Gebiete 12, 241–254 (1969). https://doi.org/10.1007/BF00534843
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DOI: https://doi.org/10.1007/BF00534843