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Birth and death on a Brownian flow: a Feller semigroup and its generator and a martingale problem

Published online by Cambridge University Press:  14 July 2016

Michael J. Phelan*
Affiliation:
University of Pennsylvania
*
Postal address: Department of Statistics, The Wharton School of the University of Pennsylvania, 3000 Steinberg Hall-Dietrich Hall, Philadelphia, PA 19104–6302, USA.

Abstract

We consider a system of particles in birth and death on a stochastic flow. The system includes a particle process tracking the spatial configuration of live particles on the flow. The particle process is a Markov process on the space of bounded counting measures. We show that its transition semigroup is a Feller semigroup and exhibit its pregenerator. The pregenerator defines a martingale problem. We show that the particle process solves the problem uniquely.

Type
Research Papers
Copyright
Copyright © Applied Probability Trust 1996 

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References

ÇlInlar, E. and Kao, J. S. (1992a) Particle systems on flows. Applied Stochastic Models and Data Analysis. Wiley, New York.Google Scholar
ÇlInlar, E. and Kao, J. S. (1992b) Birth and death on flows. Diffusion Processes and Related Problems in Analysis, II: Stochastic Flows. ed. Pinsky, M. A. and Wihstutz, V. pp. 121137. Birkhauser, Boston.Google Scholar
CONWAY, J. (1985) A Course in Functional Analysis. Springer, New York.Google Scholar
Ethier, S. N. and Kurtz, T. G. (1986) Markov Processes: Characterization and Convergence. Wiley, New York.Google Scholar
Jacod, J. and Shiryaev, A. N. (1987) Limit Theorems for Stochastic Processes. Springer, New York.Google Scholar
Kallenberg, O. (1986) Random Measures. 4th edn. Academic Press, New York.Google Scholar
Kunita, H. (1990) Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, Cambridge.Google Scholar
Métivier, M. (1982) Semimartingales: A Course in Stochastic Processes. Walter de Gruyter, New York.Google Scholar
Royden, H. L. (1968) Real Analysis. 2nd edn. MacMillan, New York.Google Scholar
Sharpe, M. J. (1988) General Theory of Markov Processes. Academic Press, New York.Google Scholar
Shiryaev, A. N. (1984) Probability. Springer, New York.Google Scholar
Treves, F. (1967) Topological Vector Spaces, Distributions and Kernels. Academic Press, New York.Google Scholar