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3 results on '"Semigroup of kernels"'

1. Additive Kernels and Integral Representation of Potentials.

Author

Hmissi, Farida and Hmissi, Mohamed

Abstract

Let P=( P t) t>0 be a submarkovian semigroup of kernels on a measurable space ( X,ℬ). An additive kernel of P is a kernel K from X into ]0,∞[ such that P t K( x, A)= K( x, A+ t) for every t>0, x∋ X and every Borel subset A of ]0,∞[. It is proved in this paper that for every potential f of P, there exits an additive kernel K of P, unique (up to equivalence) such that f= K1=∫0 ∞ K(⋅,d t). This result is already well known for regular potentials of right processes. If U=( U p) p>0 is a sub-Markovian resolvent of kernels on ( X,ℬ), we give a notion of additive kernel of U and we prove a similar integral representation of potentials of U. [ABSTRACT FROM AUTHOR]

Published
2001
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2. On Markov Processes with Random Starting Time

Author

Talma Leviatan

Subjects
Cauchy problem, Statistics and Probability, Pure mathematics, Semigroup of kernels, Markov kernel, Markov chain, Variable-order Markov model, Markov process, Markov model, Time reversibility, Combinatorics, symbols.namesake, 60.69, Markov renewal process, symbols, 60.60, Perturbation theory for Markov processes, Markov property, Statistics, Probability and Uncertainty, Mathematics
Abstract

The paper deals with Markov processes which have both random starting and terminal times. Such processes were suggested by G. A. Hunt, were constructed by L. L. Helms (under the name Markov processes with creation and annihilation) and were treated also by M. Nagasawa and the author. The paper contains a new existence proof by a way of constructing such a process from its given associated semigroup of kernels $\tilde{P}_t, t \geqq 0$, and its (Markov) transition function. This construction is more general than that given by L. L. Helms (in terms of the Markov transition function and the creation measure) and is also more convenient as far as perturbation theory of Markov processes is concerned. Indeed more general relations between this theory and creation of mass processes are established. Finally an application to solving the Cauchy problem in partial differential equations is indicated.

Published
1973

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