Summary
We consider two classes of measure-valued diffusion processes; measure-valued branching diffusions and Fleming-Viot diffusion models. When the basic space is R 1, and the drift operator is a fractional Laplacian of order 1<α≦2, we derive stochastic partial differential equations based on a space-time white noise for these two processes. The former is the expected one by Dawson, but the latter is a new type of stochastic partial differential equation.
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Dawson, D.A.: Stochastic evolution equations and related measure processes. J. Multivariate Anal. 5, 1–52 (1975)
Dawson, D.A.: The critical measure diffusion process. Z. Wahrscheinlichkeitstheor. Verw. Geb. 40, 125–145 (1977)
Dawson, D.A., Hochberg, K.J.: The carrying dimension of a stochastis measure diffusion. Ann. Probab. 7, 693–703 (1979)
Dawson, D.A., Hochberg, K.J.: Wandering random measures in the Fleming-Viot model. Ann. Probab. 10, 554–580 (1982)
Fleischmann, K.: Critical behavior of measure-valued processes. Preprint 1986
Fleming, W.H., Viot, M.: Some measure-valued Markov Processes in population genetics theory. Indiana Univ. Math. J. 28, 817–843 (1979)
Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Tokyo: North-Holland/Kodansha 1981
Iscoe, I.: A weighted occupation time for a class of measure-valued branching processes. Probab. Th. Rel. Fields 71, 85–116 (1986)
Perkins, E.A.: A space-time property of a class of measure-valued branching diffusions. Trans. Am. Math. Soc. (in press)
Reimers, M.A.: Hyper-finite methods for multi-dimensional stochastic processes. PhD Dissertation. University of British Columbia (1986)
Roelly-Coppoletta, S.: A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics 17, 43–65 (1986)
Shiga, T.: A certain class of infinite dimensional diffusion processes arising in population genetics. J. Math. Soc. Japan 30, 17–25 (1987)
Strook, D.W., Varadhan, S.R.S.: Multidimensional diffusion processes. Berlin Heidelberg New York: Springer, 1979
Walsh, J.B.: An introduction to stochastic partial differential equations. Lect. Notes Math., vol. 1180, pp. 265–439. Berlin Heidelberg New York: Springer 1986
Watanabe, A.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167 (1968)
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Konno, N., Shiga, T. Stochastic partial differential equations for some measure-valued diffusions. Probab. Th. Rel. Fields 79, 201–225 (1988). https://doi.org/10.1007/BF00320919
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DOI: https://doi.org/10.1007/BF00320919