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Measure-valued Markov processes

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Book cover Ecole d'Eté de Probabilités de Saint-Flour XXI - 1991

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References

  • R.J. Adler and M. Lewin (1991) An evolution equation for the intersection local times of superdiffusions, Stochastic Analysis, Cambridge Univ. Press, 1–22.

    Google Scholar 

  • R.J. Adler and M. Lewin (1992). Local time and Tanaka Formulae for super-Brownian motion and super stable processes, Stoch. Proc. Appl., 41, 45–68.

    Article  MathSciNet  MATH  Google Scholar 

  • D.J. Aldous (1985). Exchangeability and related topics, Lecture Notes in Math. 1117, 2–198.

    MathSciNet  MATH  Google Scholar 

  • D.J. Aldous (1991a). Asymptotic fringe distributions for general families of random trees, Ann. Appl. Prob. 1, 228–266.

    Article  MathSciNet  MATH  Google Scholar 

  • D.J. Aldous (1991b). The continuum random tree II: an overview, in Stochastic Analysis, ed. M.T. Barlow and N.H. Bingham, 23–70, Cambridge Univ. Press.

    Google Scholar 

  • D.J. Aldous (1992). The continuum random tree III, preprint.

    Google Scholar 

  • K.B. Athreya (1969). On a characteristic property of Pólya's urn, Stud. Sci. Math. Hung., 4, 31–35.

    MathSciNet  MATH  Google Scholar 

  • K.B. Athreya and P.E. Ney (1977). Branching Processes, Springer-Verlag.

    Google Scholar 

  • P. Baras, M. Pierre (1984). Singularités éliminables pour des équations semi-linéaires, Ann. INst. Fourier 34, 185–206.

    Article  MathSciNet  MATH  Google Scholar 

  • M.T. Barlow, S.N. Evans and E.A. Perkins (1991) Collision local times and measure-valued processes, Can. J. Math. 43, 897–938.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Berg, J.P.R. Christensen and P. Ressel (1984). Harmonic Analysis on Semigroups, Springer-Verlag.

    Google Scholar 

  • P. Billingsley (1968). Convergence of Probability Measures, John Wiley.

    Google Scholar 

  • D. Blackwell and D.G. Kendall (1964). The Martin boundary for Polya's urn scheme and an application to stochastic population growth, J. Appl. Prob. 1, 284–296.

    MathSciNet  MATH  Google Scholar 

  • D. Blackwell and J.B. MacQueen (1973). Ferguson distributions via Pólya urn schemes, Ann. Stat. 1, 353–355.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Blackwell and L.E. Dubins (1983). An extension of Skorohod's almost sure representation theorem, Proc. A.M.S. 89, 691–692.

    MathSciNet  MATH  Google Scholar 

  • R.M. Blumenthal and R.K. Getoor (1968). Markov Processes and Potential Theory, Academic Press, New York.

    MATH  Google Scholar 

  • V.S. Borkar (1984). Evolution of interacting particles in a Brownian medium, Stochastics 14, 33–79.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Bose and I. Kaj (1991a) Diffusion approximation for an age-structured population, LRSP Tech. Report 148, Carleton Univ.

    Google Scholar 

  • A. Bose and I. Kaj (1991b). Measure-valued age-structured processes, LRSP Tech. Report 161, Carleton Univ.

    Google Scholar 

  • L. Breiman (1968). Probability, Addison-Wesley.

    Google Scholar 

  • H. Brezis, L.A. Peletier and D. Terman (1986). A very singular solution of the heat equation with absorption, Arch. Rational Mech. Anal. 95, 185–209.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Brezis and L. Veron (1980). Removable singularities of some nonlinear elliptic equations, Arch. Rational Mech. Anal. 75, 1–6.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Brezis and A. Friedman (1983) Nonlinear parabolic equations involving measures as initial conditions, J. Math. pures et appl. 62, 73–97.

    MathSciNet  MATH  Google Scholar 

  • O.G. Bulycheva and A.D. Vent-tsel' (1989). On the differentiability of expectations of functionals of a Wiener process, Th. Prob. Appl. 34, 509–512.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Cannings (1974). The latent roots of certain Markov chains arising in genetics: A new approach 1. Haploid models. Adv. Appl. Probab. 6, 260–290.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Chauvin (1986a). Arbres et processus de Bellman-Harris, Ann. Inst. Henri Poincaré 22, 209–232.

    MathSciNet  MATH  Google Scholar 

  • B. Chauvin (1986b). Sur la propriéte de branchement, Ann. Inst. Henri Poincaré 22, 233–236.

    MathSciNet  MATH  Google Scholar 

  • B. Chauvin, A. Rouault and A. Wakolbinger (1989). Growing conditioned trees, Stoch. Proc. Appl. 39, 117–130.

    Article  MathSciNet  MATH  Google Scholar 

  • P.L. Chow (1976). Function space differential equations associated with a stochastic partial differential equation, Indiana Univ. Math. J. 25, 609–627.

    Article  MathSciNet  MATH  Google Scholar 

  • P.L. Chow (1978). Stochastic partial differential equations in turbulence related problems. In Probabilistic Analysis and Related Topics, Vol. 1, Academic Press.

    Google Scholar 

  • K.L. Chung, P. Erdos and T. Sirao (1959). On the Lipschitz condition for Brownian motions, J. Math. Soc. Japan 11, 263–274.

    Article  MathSciNet  MATH  Google Scholar 

  • Z. Ciesielski and S.J. Taylor (1962). First passage times and sojourn times for Brownian motion and exact Hausdorff measure of the sample path, Trans. Amer. Math. Soc. 103, 434–450.

    Article  MathSciNet  MATH  Google Scholar 

  • J.T. Cox and D. Griffeath (1985). Occupation times for critical branching Brownian motions, Ann. Probab. 13, 1108–1132.

    Article  MathSciNet  MATH  Google Scholar 

  • J.T. Cox and D. Griffeath (1987). Recent results on the stepping stone model,in Percolation Theory and Ergodic Theory of Infinite Particle Systems, 73–83, IMA Volume 8, ed. H. Kesten, Springer-Verlag.

    Google Scholar 

  • J.T. Cox and D. Griffeath (1990). Mean-field asymptotics for the planar stepping stone model, Proc. London Math. Soc. 61, 189–208.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F. Crow and M. Kimura (1970). An Introduction to Population Genetics, Burgess.

    Google Scholar 

  • C. Cutler (1984a). Some measure-theoretic and topological results for measure-valued and set-valued stochastic processes, Ph.D. Thesis, Carleton University.

    Google Scholar 

  • C. Cutler (1984b). A Lebesgue decomposition theorem for random measures and random measure processes, Tech Report 23, LRSP, Carleton University.

    Google Scholar 

  • Dai Yonglong (1982). On absolute continuity and singularity of random measures (in Chinese), Chinese Annals of Mathematics 3, 241–246.

    MATH  Google Scholar 

  • Yu. Dalecky and S. Fomin (1991). The measures and differential equations in infinite dimensional spaces, Kluwer.

    Google Scholar 

  • D.J. Daley and D. Vere-Jones (1988). An Introduction to the Theory of Point Processes, Springer-Verlag.

    Google Scholar 

  • D.A. Dawson (1975). Stochastic evolution equations and related measure-valued processes, J. Multivariate Analysis 5, 1–52.

    Article  MATH  Google Scholar 

  • D.A. Dawson (1977). The critical measure diffusion, Z. Wahr. verw Geb. 40, 125–145.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson (1978a). Geostochastic calculus, Canadian Journal of Statistics 6, 143–168.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson (1978b). Limit theorems for interaction free geostochastic systems, Colloquia Math. Soc. J. Bolyai, 24, 27–47.

    MathSciNet  Google Scholar 

  • D.A. Dawson (1986a). Measure-valued stochastic processes: construction, qualitative behavior and stochastic geometry, Proc. Workshop on Spatial Stochastic Models, Lecture Notes in Mathematics 1212, 69–93, Springer-Verlag.

    Google Scholar 

  • D.A. Dawson (1986b). Stochastic ensembles and hierarchies, Lecture Notes in Mathematics 1203, 20–37, Springer-Verlag.

    Google Scholar 

  • D.A. Dawson (1992). Infinitely Divisible Random Measures and Superprocesses, in Proc. 1990 Workshop on Stochastic Analysis and Related Topics, Silivri, Turkey.

    Google Scholar 

  • D.A. Dawson and K. Fleischmann (1991) Critical branching in a highly fluctuating random medium, Probab. Theory Rel. Fields, 90, 241–274.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and K. Fleischmann (1992). Diffusion and reaction caused by point catalysts, SIAM J. Appl. Math. 52, 163–180.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson, K. Fleischmann, R.D. Foley and L.A. Peletier (1986). A critical measure-valued branching process with infinite mean, Stoch. Anal. Appl. 4, 117–129.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson, K. Fleischmann, and L.G. Gorostiza, (1989). Stable hydrodynamic limit fluctuations of a critical branching particle system, Ann. Probab. 17, 1083–1117.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson, K. Fleischmann and S. Roelly (1991). Absolute continuity of the measure states in a branching model with catalysts, Seminar on Stochastic processes 1990, Birkhäuser, 117–160.

    Google Scholar 

  • D.A. Dawson and K.J. Hochberg (1979). The carrying dimmension of a stochastic measure diffusion, Ann. Prob. 7, 693–703.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and K.J. Hochberg (1982). Wandering random measures in the Fleming-Viot model, Ann. Prob. 10, 554–580.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and K.J. Hochberg (1985). Function-valued duals for measure-valued processes with applications, Contemporary Mathematics 41, 55–69.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson, K.J. Hochberg and Y. Wu (1990). Multilevel branching systems, in Proc. Bielefeld Encounters in Mathematics and Physics 1989, World Scientific, 93–107.

    Google Scholar 

  • D.A. Dawson and K.J. Hochberg (1991). A multilevel branching model, Adv. Appl. Prob. 23, 701–715.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson, I. Iscoe and E.A. Perkins (1989). Super-Brownian motion: path properties and hitting probabilities, Probab. Th. Rel. Fields 83, 135–205.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and B.G. Ivanoff (1978). Branching diffusions and random measures. In Stochastic Processes, ed. A. Joffe and P. Ney, 61–104, Dekker, New York.

    Google Scholar 

  • D.A. Dawson and T.G. Kurtz (1982). Applications of duality to measure-valued processes, Lecture Notes in Control and Inform. Sci. 42, 177–191.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and P. March (1992). In preparation.

    Google Scholar 

  • D.A. Dawson and E.A. Perkins (1991). Historical processes, Memoirs of the American Mathematical Society 93, no. 454.

    Google Scholar 

  • D.A. Dawson and H. Salehi (1980). Spatially homogeneous random evolutions, J. Mult. Anal. 10, 141–180.

    Article  MathSciNet  MATH  Google Scholar 

  • D.A. Dawson and V. Vinogradov (1992a). Almost sure path properties of (2, d,β) super-processes, LRSP Tech. Report 195.

    Google Scholar 

  • D.A. Dawson and V. Vinogradov (1992b). Mutual singularity of genealogical structures of Fleming-Viot and continuous branching processes, LRSP Tech Report 204.

    Google Scholar 

  • C. Dellacherie and P.A. Meyer (1976). Probabilités et potentiel, Hermann, Vol. I 1976, Vol. II 1980, Vol. III 1983, Vol. IV 1987.

    Google Scholar 

  • A. De Masi and E. Presutti (1991). Mathematical Methods for Hydrodynamic Limits, Lecture Notes in Mathematics 1501, Springer Verlag.

    Google Scholar 

  • P. Donnelly (1984). The transient behavior of the Moran model in population genetics, Math. Proc. Camb. Phil Soc. 95, 349–358.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Donnelly (1985). Dual processes and an invariance result for exchangeable models in population genetics, J. Math. Biol.

    Google Scholar 

  • P. Donnelly (1986) Partition structures, Polya urns, the Ewens sampling formula and the ages of alleles, Theor. Pop. Biol. 30, 271–288.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Donnelly (1991). Weak convergence to a Markov chain with an entrance boundary: ancestral processes in population genetics, Ann. Probab. 19, 1102–1117.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Donnelly and P. Joyce (1992). Weak convergence of population genealogical processes to the coalescent with ages, Ann. Prob. 20, 322–341.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Donnelly and T.G. Kurtz (1992) The Fleming Viot measure-valued diffusion as an interactive particle system, preprint.

    Google Scholar 

  • P. Donnelly and S. Tavaré (1986). The ages of alleles and a coalescent, Adv. Appl. Prob. 18, 1–19.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Donnelly and S. Tavaré (1987). The population genealogy of the infinitely-many neutral alleles model, J. Math. Biol. 25, 381–391.

    Article  MathSciNet  MATH  Google Scholar 

  • J.L. Doob (1984). Classical Potential Theory and Its Probabilistic Counterpart, Springer-Verlag.

    Google Scholar 

  • R. Durrett (1978). The genealogy of critical branching processes, Stoch. Proc. Appl. 8, 101–116.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Durrett (1988). Lecture Notes on Particle Systems and Percolation, Wadsworth and Brooks/Cole.

    Google Scholar 

  • E.B. Dynkin (1965). Markov Processes, Volumes I and II, Springer-Verlag.

    Google Scholar 

  • E.B. Dynkin, (1988). Representation for functionals of superprocesses by multiple stochastic integrals, with applications to self intersection local times, Astérisque 157–158, 147–171.

    MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin, (1989a). Superprocesses and their linear additive functionals, Trans. Amer. Math. Soc., 314, 255–282.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin, (1989b). Regular transition functions and regular superprocesses, Trans. Amer. Math. Soc., 316, 623–634.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin, (1989c). Three classes of infinite dimensional diffusions, J. Funct. Anal. 86, 75–110.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin, (1991a). Branching particle systems and superprocesses, Ann. Probab., 19, 1157–1194.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin (1991b), Path processes and historical superprocesses, Probab. Th. Rel. Fields 90, 1–36.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin (1991c) A probabilistic approach to one class of nonlinear differential equations, Probab. Th. Rel. Fields 89, 89–115.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin (1991d) Additive functionals of superdiffusion processes, in Random Walks, Brownian Motion and Interacting Particle Systems, A Festschrift in Honor of Frank Spitzer, 269–282, R. Durrett and H. Kesten, eds., Birkhäuser.

    Google Scholar 

  • E.B. Dynkin (1992a) Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20, 942–962.

    Article  MathSciNet  MATH  Google Scholar 

  • E.B. Dynkin (1992b). Superprocesses and partial differential equations, (1991 Wald Memorial Lectures).

    Google Scholar 

  • E.B. Dynkin, S.E. Kuznetsov and A.V. Skorohod (1992). Branching measure-valued processes, preprint.

    Google Scholar 

  • N. El Karoui (1985). Non-linear evolution equations and functionals of measurevalued branching processes. In Stochastic Differential Systems, ed. M. Metivier and E. Pardoux, Lect. Notes Control and Inf. Sci. 69, 25–34., Springer-Verlag.

    Google Scholar 

  • N. El Karoui and S. Roelly (1991). Proprietes de martingales, explosion et representation de Lévy-Khinchine d'une classe de processus de branchement à valeurs mesures, Stoch. Proc. Appl. 38, 239–266.

    Article  MATH  Google Scholar 

  • N. El Karoui and S. Méléard (1990) Martingale measures and stochastic calculus, Prob. Th. Rel Fields. 84, 83–101.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Etheridge and P. March (1991) A note on superprocesses, Probab. Theory Rel. Fields, 89, 141–147.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier (1976). A class of degenerate diffusion processes occurring in population genetics, Comm. Pure Appl. Math. 29, 483–493.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier (1979). Limit theorems for absorption times of genetic models, Ann. Prob. 7, 622–638.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier (1981). A class of infinite-dimensional diffusions occurring in population genetics, Indiana Univ. Math. J. 30, 925–935.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier (1988). The infinitely-many-neutral-alleles diffusion model with ages, Adv. Appl. Prob. 22, 1–24.

    MathSciNet  MATH  Google Scholar 

  • S.N. Ethier (1990a) On the stationary distribution of the neutral one-locus diffusion model in population genetics, Ann. Appl. Prob. 2, 24–35.

    Article  MathSciNet  Google Scholar 

  • S.N. Ethier (1990b) The distribution of the frequencies of age-ordered alleles in a diffusion model, Adv. Appl. Prob. 22, 519–532.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier and R.C. Griffiths (1987). The infinitely many sites model as a measure-valued diffusion, Ann. Prob. 15, 515–545.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier and R.C. Griffiths (1988). The two locus infinitely many neutral alleles diffusion model, preprint.

    Google Scholar 

  • S.N. Ethier and R.C. Griffiths (1990) The neutral two locus model as a measure-valued diffusion, Adv. Appl. Prob.

    Google Scholar 

  • S.N. Ethier and R.C. Griffiths (1992) The transition function of a Fleming-Viot process, preprint.

    Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1981). The infinitely many neutral alleles diffusion model, Adv. Appl. Prob. 13, 429–452.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1985). Markov processes: characterization and convergence, Wiley.

    Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1987). The infinitely many alleles model with selection as a measure-valued diffusion, Lecture Notes in Biomathematics 70, 72–86.

    Article  MathSciNet  MATH  Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1990a) Coupling and ergodic theorems for Fleming-Viot processes, preprint.

    Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1990b) Convergence to Fleming-Viot processes in the weak atomic topology, Stochatic Proc. Appl. to appear.

    Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1992a) On the stationary distribution of the neutral diffusion model in population genetics, Ann. Appl. Prob. 2.

    Google Scholar 

  • S.N. Ethier and T.G. Kurtz (1992b). Fleming-Viot processes in population genetics, preprint.

    Google Scholar 

  • S.N. Evans (1990). The entrance space of a measure-valued Markov branching process conditioned on non-extinction. Tech. Rept. 230, Dept. of Stat., Univ. of California at Berkeley.

    Google Scholar 

  • S.N. Evans (1991) Trapping a measure-valued branching process conditioned on non-extinction, Ann. Inst. Henri Poincaré 27, 215–220.

    MATH  Google Scholar 

  • S.N. Evans (1992) The entrance space of a measure-valued Markov branching proces conditioned on non-extinction, Can. Math. Bull., to appear.

    Google Scholar 

  • S. Evans and E. Perkins (1990). Measure-valued Markov branching processes conditioned on non-extinction, Israel J. Math., 71, 329–337.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Evans and E. Perkins (1991). Absolute continuity results for superprocesses with some applications, Trans. Amer. Math. Soc., 325, 661–681.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Evans and E.A. Perkins (1992). Measure-valued branching diffusions with singular interaction, preprint.

    Google Scholar 

  • W.J. Ewens (1979). Mathematical Population Genetics, Springer-Verlag.

    Google Scholar 

  • K.J. Falconer (1985). The Geometry of Fractal Sets, Cambridge Univ. Press.

    Google Scholar 

  • H. Federer (1969). Geometric measure theory, Springer-Verlag.

    Google Scholar 

  • P.D. Feigen and R.L. Tweedie (1989). Linear functionals and Markov chains associated with Dirichlet processes, Math. Proc. Camb. Phil. Soc. 105, 579–585.

    Article  MathSciNet  MATH  Google Scholar 

  • W. Feller (1951). Diffusion processes in genetics, Proc. Second Berkeley Symp., Univ. of Calif. Press, Berkeley, 227–246.

    Google Scholar 

  • T.S. Ferguson (1973). A Bayesian analysis of some nonparametric problems, Ann. Stat. 1, 209–230.

    Article  MathSciNet  MATH  Google Scholar 

  • X. Fernique (199.) Fonctions aléatoires à valeurs dans les espaces lusiniens, Expositiones Math.

    Google Scholar 

  • R.A. Fisher (1958). The genetic theory of natural selection, Dover.

    Google Scholar 

  • P.J. Fitzsimmons (1988). Construction and regularity of measure-valued branching processes, Israel J. Math. 64, 337–361.

    Article  MathSciNet  MATH  Google Scholar 

  • P.J. Fitzsimmons (1991). Correction to Construction and regularity of measure-valued branching processes, Israel J. Math. 73, 127.

    MathSciNet  MATH  Google Scholar 

  • P.J. Fitzsimmons (1992). On the martingale problem for measure-valued Markov branching processes, in Seminar on Stochastic Processes, 1991, E. Cinlar, K.L. Chung and M.J. Sharpe, eds., Birkhäuser.

    Google Scholar 

  • K. Fleischmann (1988). Critical behavior of some measure-valued processes, Math. Nachr. 135, 131–147.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Fleischmann and J. Gärtner (1986). Occupation time process at a critical point, Math. Nachr. 125, 275–290.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Fleischmann and U. Prehn (1974). Ein Grenzwertsatz für subkritische Verzweigungsprozesse mit endlich vielen Typen von Teilchen, Math. Nachr. 64, 357–362.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Fleischmann and U. Prehn (1975). Subkritische räumlich homogene Verzweigungsprozesse, Math. Nachr. 70, 231–250.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Fleischmann and R. Sigmund-Schultze (1977). The structure of reduced critical Galton-Watson processes, Math. Nachr. 74, 233–241.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Fleischmann and R. Sigmund-Schultze (1978). An invariance principle for reduced family trees of critically spatially homogeneous branching processes (with discussion), Serdica Bulg. Math. 4, 11–134.

    Google Scholar 

  • W.H. Fleming and M. Viot (1979). Some measure-valued Markov processes in population genetics theory, Indiana Univ. Math. J. 28, 817–843.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Gärtner (1988). On the McKean-Vlasov limit for interacting diffusions, Math. Nachr. 137, 197–248.

    Article  MathSciNet  MATH  Google Scholar 

  • R.K. Getoor (1974). Markov processes: Ray processes and right processes, Lecture Notes in Math. 440, Springer-Verlag.

    Google Scholar 

  • R.K. Getoor (1975). On the construction of kernels, Sem. de Prob. IX., Lecture Notes in Mathematics 465, 441–463, Springer-Verlag.

    Google Scholar 

  • A. Gmira, L. Veron (1984). Large time behavior of the solutions of a semilinear parabolic equation in ℝN, J. Diff. Equations 53, 258–276.

    Article  MathSciNet  MATH  Google Scholar 

  • D.E. Goldberg (1989). Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley.

    Google Scholar 

  • L.G. Gorostiza (1981). Limites gaussiennes pour les champs aléatoires ramifiés supercritiques, Colloque CNRS Aspects statistiques et aspects physiques des processus gaussiens, 385–398.

    Google Scholar 

  • L.G. Gorostiza and J.A. López-Mimbela (1990). The multitype measure branching process, Adv. Appl. Prob. 22, 49–67.

    Article  MathSciNet  MATH  Google Scholar 

  • L.G. Gorostiza and J. A. López-Mimbela (1992). A convergence criterion for measure-valued processes, and application to continuous superprocesses, Prog. in Probab., Birkhäuser, to appear.

    Google Scholar 

  • L.G. Gorostiza and S. Roelly-Coppoletta (1990) Some properties of the multitype measure branching process, Stoch. Proc. Appl. 37, 259–274.

    Article  MathSciNet  MATH  Google Scholar 

  • L.G. Gorostiza, S. Roelly-Coppoletta and A. Wakolbinger (1990). Sur la persistence du processus de Dawson-Watanabe stable; intervention del la limite en temps et de la renormalization, Sém. Probab. XXIV, Lecture Notes in Math. 1426. 275–281.

    MathSciNet  MATH  Google Scholar 

  • L.G. Gorostiza, S. Roelly and A. Wakolbinger (1992) Persistence of critical multitype particle and measure branching processes, Prob. Th. Rel. Fields.

    Google Scholar 

  • L.G. Gorostiza and A. Wakolbinger (1991). Persistence criteria for a class of critical branching particle systems in continuous time, Ann. Probab. 19, 266–288.

    Article  MathSciNet  MATH  Google Scholar 

  • L.G. Gorostiza and A. Wakolbinger (1992). Convergence to equilibrium of critical branching particle systems and superprocesses, and related nonlinear partial differential equations, Acta Appl. Math., to appear.

    Google Scholar 

  • R.C. Griffiths (1979) A transition density expansion for a multi-allele diffusion model, Adv. Appl. Prob. 11, 310–325.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Gyöngy and E. Pardoux (1991). On quasi-linear stochastic partial differential equations, Probab. Th. Rel. Fields.

    Google Scholar 

  • K. Handa (1990) A measure-valued diffusion process describing the stepping stone model with infinitely many alleles, Stoch. Proc. Appl. 36, 269–296.

    Article  MathSciNet  MATH  Google Scholar 

  • T.E. Harris (1963). The Theory of Branching Processes, Springer-Verlag.

    Google Scholar 

  • K.J. Hochberg (1991) Measure-valued processes: techniques and applications. In Selected Proc. Sheffield Symp. Appl. Probab. IMS Lecture Notes-Monograph Series 18, 212–235.

    Article  MathSciNet  MATH  Google Scholar 

  • K.J. Hochberg (1986). Stochastic population theory: Mathematical evolution of a genetical model, in New Directions in Applied and Computational Mathematics, 101–115, Springer.

    Google Scholar 

  • R.A. Holley and D.W. Stroock (1978). Generalized Ornstein-Uhlenbeck processes and infinite particle branching Brownian motion, Publ. R.I.M.S. Kyoto Univ. 14, 741–788.

    Article  MathSciNet  MATH  Google Scholar 

  • R.A. Holley and D.W. Stroock (1979). Central limit phenomena of various interacting systems, Ann. Math. 110, 333–393.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Holley and T. Liggett (1975). Ergodic theorems for weakly interacting systems and the voter model, Ann. Prob. 3, 643–663.

    Article  MathSciNet  MATH  Google Scholar 

  • F.M. Hoppe (1987). The sampling theory of neutral alleles and an urn model in population genetics, J. Math. Biol. 25, 123–159.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Ikeda, M. Nagasawa and S. Watanabe (1968), (1969). Branching Markov processes I,II,III, J. Math. Kyoto Univ. 8, 233–278, 9, 95–160.

    MathSciNet  MATH  Google Scholar 

  • N. Ikeda and S. Watanabe (1981). Stochastic differential equations and diffusion processes, North Holland.

    Google Scholar 

  • I. Iscoe (1980). The man-hour process associated with measure-valued branching random motions ind, Ph.D. thesis, Carleton University.

    Google Scholar 

  • I. Iscoe (1986a). A weighted occupation time for a class of measure-valued critical branching Brownian motion, Probab. Th. Rel. Fields 71, 85–116.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Iscoe (1986b). Ergodic theory and a local occupation time for measure-valued branching processes, Stochastics 18, 197–143.

    Article  MathSciNet  MATH  Google Scholar 

  • I. Iscoe (1988). On the supports of measure-valued critical branching Brownian motion, Ann. Prob. 16, 200–221.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Itatsu (1981). Equilibrium measures of the stepping stone model in population genetics, Nagoya Math. J. 83, 37–51.

    MathSciNet  MATH  Google Scholar 

  • K. Itô and H.P. McKean (1965). Diffusion processes and their sample paths, Springer-Verlag.

    Google Scholar 

  • K. Itô (1984). Foundations of stochastic differential equations in infinite dimensional space, SIAM.

    Google Scholar 

  • B.G. Ivanoff (1981). The multitype branching diffusion, J. Mult. Anal. 11, 289–318.

    Article  MathSciNet  MATH  Google Scholar 

  • B.G. Ivanoff (1989). The multitype branching random walk: temporal and spatial limit theorems, preprint.

    Google Scholar 

  • K. Iwata (1987). An infinite dimensional stochastic differential equation with state space C(ℝ), Prob. Th. Rel. Fields 74, 141–159.

    Article  MathSciNet  Google Scholar 

  • J. Jacod (1979). Calcul Stochastiques et Problèmes de Martingales, LNM 714, Springer-Verlag.

    Google Scholar 

  • J. Jacod and A.N. Shiryaev (1987). Limit theorems for stochastic processes, Springer-Verlag.

    Google Scholar 

  • P. Jagers (1974). Aspects of random measures and point processes. In Advances in Probability, P. Ney and S. Port, eds., M. Dekker, 179–238.

    Google Scholar 

  • P. Jagers (1975). Branching processes with biological applications, Wiley.

    Google Scholar 

  • P. Jagers and O. Nerman (1984). The growth and composition of branching processes, Adv. Appl. Prob. 16, 221–259.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Jakubowski (1986). On the Skorohod topology, Ann. Inst. H. Poincaré B22, 263–285.

    MathSciNet  MATH  Google Scholar 

  • M. Jirina (1958). Stochastic branching processes with continuous state space, Czechoslovak Math. J. 8., 292–313.

    MathSciNet  MATH  Google Scholar 

  • M. Jirina (1964). Branching processes with measure-valued states, In. Trans. Third Prague Conf. on Inf. Th., 333–357.

    Google Scholar 

  • A. Joffe and M. Métivier (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. Appl. Prob. 18, 20–65.

    Article  MathSciNet  MATH  Google Scholar 

  • N.L. Johnson and S. Kotz (1977). Urn Models and Their Applications, Wiley.

    Google Scholar 

  • O. Kallenberg (1977). Stability of critical cluster fields, Math. Nachr. 77, 7–43.

    Article  MathSciNet  MATH  Google Scholar 

  • O. Kallenberg (1983). Random measures, 3rd ed., Akademie Verlag and Academic Press.

    Google Scholar 

  • N.L. Kaplan, T. Darden and R.R. Hudson (1988) The coalescent process in models with selction, Genetics 120, 819–829.

    Google Scholar 

  • K. Kawazu and S. Watanabe (1971). Branching processes with immigration and related limit theorems, Th. Prob. Appl. 26, 36–54.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Kimura (1983a). The neutral theory of molecular evolution, Cambridge Univ. Press.

    Google Scholar 

  • M. Kimura (1983b). Diffusion model of intergroup selection, with special reference to evolution of an altruistic character, Proc. Nat. Acad. Sci. USA 80, 6317–6321.

    Article  MATH  Google Scholar 

  • J.F.C. Kingman (1975). Random discrete distributions, J.R. Statist. Soc. B37, 1–22.

    MathSciNet  MATH  Google Scholar 

  • J.F.C. Kingman (1978). Uses of exchangeability, Ann. Probab. 6, 183–197.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F.C. Kingman (1980) The mathematics of Genetic Diversity, CBMS Regional Conf. Series in Appl. Math. 34, SIAM.

    Google Scholar 

  • J.F.C. Kingman (1982a). The coalescent, Stoch. Proc. Appl. 13, 235–248.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F.C. Kingman (1982b) On the genealogy of large populations, J. Appl. Prob. 19A, 27–43.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F.C. Kingman (1982c). Exchangeability and the evolution of large populations, in Exchangeability in Probability and Statistics, eds. G. Koch and F. Spizzichino, 97–112, North Holland.

    Google Scholar 

  • F. Knight (1981). Essentials of Brownian Motion and Diffusion, Amer. Math. Soc., Providence.

    Google Scholar 

  • N. Konno and T. Shiga (1988). Stochastic differential equations for some measurevalued diffusions, Prob. Th. Rel Fields 79, 201–225.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Kotelenez (1988). High density limit theorems for nonlinear chemical reactions with diffusion, Probab. Th. Rel. Fields 78, 11–37.

    Article  MathSciNet  MATH  Google Scholar 

  • P. Kotelenez (1989). A class of function and density valued stochastic partial differential equations driven by space-time white noise, preprint.

    Google Scholar 

  • S. Krone (1990) Local times for superdiffusions (Abstract), Stoch. Proc. Appl. 35, 199–200.

    Google Scholar 

  • N.V. Krylov and B.L. Rozovskii (1981). Stochastic evolution equations, J. Soviet Math. (Itogi Nauki i Techniki 14), 1233–1277.

    Article  MATH  Google Scholar 

  • H. Kunita (1986). Stochastic flows and applications, Tata Institute and Springer-Verlag.

    Google Scholar 

  • H. Kunita (1990). Stochastic flows and stochastic differential equations, Cambridge Univ. Press.

    Google Scholar 

  • T.G. Kurtz and D. Ocone (1988). A martingale problem for conditional distributions and uniqueness for the nonlinear filtering equations, Ann. Probab.

    Google Scholar 

  • T.G. Kurtz (1981). Approximation of Population Processes, SIAM.

    Google Scholar 

  • S.E. Kuznetsov (1984). Nonhomogeneous Markov processes, J. Soviet Math. 25, 1380–1498.

    Article  MATH  Google Scholar 

  • J. Lamperti (1967). Continuous state branching processes, Bull. Amer. Math. Soc. 73, 382–386.

    Article  MathSciNet  MATH  Google Scholar 

  • T.-Y. Lee (1990). Some limit theorems for critical branching Bessel processes and related semilinear differential equations, Probab. Th. Rel. Fields 84, 505–520.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F. Le Gall (1987). Exact Hausdorff measure of Brownian multiple points, in Seminar on Stochastic Processes, 1986, E. Cinlar, K.L. Chung and R.K. Getoor, eds., Birkhäuser.

    Google Scholar 

  • J.F. Le Gall (1989a). Marches aléatoires, mouvement brownien et processes de branchement, L.N.M. 1372, 258–274.

    Google Scholar 

  • J.F. Le Gall (1989b). Une construction de certains processus de Markov à valeurs mesures, C.R. Acad. Sci. Paris 308, Série I, 533–538.

    MathSciNet  MATH  Google Scholar 

  • J.F. Le Gall (1991a). Brownian excursions, trees and measure-valued branching processes, Ann. Probab. 19., 1399–1439.

    Article  MathSciNet  MATH  Google Scholar 

  • J.F. Le Gall (1991b). A class of path-valued Markov processes and its applications to superprocesses, preprint.

    Google Scholar 

  • Y. Le Jan (1989). Limites projectives de processus de branchement markoviens, C.R. Acad. Sci. Paris 309 Série 1, 377–381.

    MathSciNet  MATH  Google Scholar 

  • Y. Le Jan (1991). Superprocesses and projective limits of branching Markov processes, Ann. Inst. H. Poincaré 27, 91–106.

    MATH  Google Scholar 

  • C. Léonard (1986). Une loi des grands nombres pour des systèmes de diffusions avec interaction à coefficients non bornés, Ann. Inst. Henri Poincaré 22, 237–262.

    MATH  Google Scholar 

  • Z.-H. Li (1992). A note on the multitype measure branching process, Adv. Appl. Prob. 24, 496–498.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Liemant, K. Matthes and A. Wakolbinger (1988). Equilibrium Distributions of Branching Processes, Akademie-Verlag, Berlin, and Kluwer Academic Publ., Dordrecht.

    MATH  Google Scholar 

  • T.M. Liggett (1985). Interacting Particle Systems, Springer-Verlag.

    Google Scholar 

  • R. Sh. Liptser and A.N. Shiryayev (1989). Theory of Martingales, Kluwer.

    Google Scholar 

  • R.A. Littler and A.J. Good (1978). Ages, extinction times and first passage probabilities for a multiallele diffusion model with irreversible mutation, Theor. Pop. Biol. 13, 214–225.

    Article  MathSciNet  MATH  Google Scholar 

  • L. Liu and C. Mueller (1989). On the extinction of measure valued critical branching Brownian motion, Ann. Probab. 17, 1463–1465.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Marcus (1979). Stochastic diffusion on an unbounded domain, Pacific J. Math. 84, 143–153.

    Article  MathSciNet  MATH  Google Scholar 

  • G. Matheron (1975). Random sets and integral geometry, Wiley.

    Google Scholar 

  • K. Matthes, J. Kerstan and J. Mecke (1978). Infinitely Divisible Point Processes, Wiley.

    Google Scholar 

  • H.P. McKean (1969). Stochastic Integrals, Academic Press.

    Google Scholar 

  • S. Méléard and S. Roelly-Coppoletta (1990). A generalized equation for a continuous measure branching process, L.N. Math. 1390, 171–186.

    MathSciNet  MATH  Google Scholar 

  • S. Méléard and S. Roelly (1991). Discontinuous measure-valued branching processes and generalized stochastic equations, Math. Nachr. 154, 141–156.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Métivier (1982). Semimartingales, W. de Gruyter.

    Google Scholar 

  • M. Métivier and J. Pellaumail (1980). Stochastic integration, Academic Press.

    Google Scholar 

  • M. Métivier (1984). Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev, Ann. Inst. Henri Poincaré 20, 329–348.

    MathSciNet  MATH  Google Scholar 

  • M. Métivier (1985). Weak convergence of measure-valued processes using Sobolevimbedding imbedding techniques, L.N. Math. 1236, 172–183.

    Google Scholar 

  • M. Métivier (1986). Quelques problemes liés aux systèmes infini de particules et leur limites, Springer L.N.M., 426–446.

    Google Scholar 

  • M. Métivier and M. Viot (1987). On weak solutions of stochastic partial differential equations, Springer L.N.M. 1322, 139–150.

    MathSciNet  MATH  Google Scholar 

  • N.G. Meyers (1970). A theory of capacities for potentials of functions in Lebesgue classes, Math. Scand. 26, 255–292.

    MathSciNet  MATH  Google Scholar 

  • C. Mueller (1991a). Limit results for two stochastic partial differential equations, Stochastics 37, 175–199.

    MathSciNet  MATH  Google Scholar 

  • C. Mueller (1991b) On the supports of solutions to the heat equation with noise, Stochastics, 37, 225–246.

    MathSciNet  MATH  Google Scholar 

  • C. Mueller (1991). Long time existence for the heat equation with noise, Probab. Th. Rel. Fields 90, 505–518.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Mueller and E.A. Perkins (1991). The compact support property for solutions to the heat equation with noise, preprint.

    Google Scholar 

  • J. Neveu (1964). Bases Mathématiques du Calcul des Probabilités, Masson et. Cie, Paris.

    MATH  Google Scholar 

  • J. Neveu (1975). Discrete-Parameter Martingales, North-Holland.

    Google Scholar 

  • J. Neveu (1986). Arbres et processus de Galton-Watson, Ann. Inst. H. Poincaré 22, 199–207.

    MathSciNet  MATH  Google Scholar 

  • J. Neveu and J.W. Pitman (1980). The branching process in a Brownian excursion, LNM 1372, 248–257, Springer-Verlag.

    MathSciNet  MATH  Google Scholar 

  • J.M. Noble (1992). Evolution equations with random potential, private communication.

    Google Scholar 

  • M. Notohara and T. Shiga (1980). Convergence to genetically uniform state in stepping stone models of population genetics, J. Math. Biol. 10, 281–294.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Oelschläger (1989). On the derivation of reaction-diffusion equations as limit dynamics of systems of moderately interacting stochastic processes, Probab. Th. Rel. Fields 82, 565–586.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Oelschläger (1990) Limit theorems for age-structured populations, Ann. Probab. 18, 290–318.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Ohta and M. Kimura (1973). A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population, Genet. Res. 22, 201–204.

    Article  MathSciNet  Google Scholar 

  • E. Pardoux (1975). Equations aux dérivées partielles stochastiques non lineaires monotone. Etude des solutions forte de type Ito, Thèse, Univ. de Paris Sud, Orsay.

    Google Scholar 

  • K.R. Parthasarathy (1967). Probability Measures on Metric Spaces, Academic Press.

    Google Scholar 

  • A. Pazy (1983). Semigroups of linear operators and applications to partial differential equations, Springer-Verlag.

    Google Scholar 

  • E.A. Perkins (1988). A space-time property of a class of measure-valued branching diffusions, Trans. Amer. Math. Soc., 305, 743–795.

    Article  MathSciNet  MATH  Google Scholar 

  • E.A. Perkins (1989). The Hausdorff measure of the closed support of super-Brownian motion, Ann. Inst. Henri Poincaré 25, 205–224.

    MathSciNet  MATH  Google Scholar 

  • E.A. Perkins (1990). Polar sets and multiple points for super-Brownian motion, Ann. Probab. 18, 453–491.

    Article  MathSciNet  MATH  Google Scholar 

  • E.A. Perkins (1991a) On the continuity of measure-valued processes, Seminar on Stochastic Processes 1990, Birkhauser, 261–268.

    Google Scholar 

  • E.A. Perkins (1991b) Conditional Dawson-Watanabe processes and Fleming-Viot processes, Seminar in Stochastic Processes, 1991, Birkhauser, 142–155.

    Google Scholar 

  • E.A. Perkins (1992). Measure-valued branching diffusions with spatial interactions, Probab. Th. Rel. Fields, to appear.

    Google Scholar 

  • P. Priouret (1974). Processus de diffusion et equations differentielles stochastiques, Lecture Notes in Math. 390, 38–111, Springer-Verlag.

    MathSciNet  MATH  Google Scholar 

  • P. Protter (1990). Stochastic Integration and Differential Equations, Springer-Verlag.

    Google Scholar 

  • M. Reimers (1986). Hyper-finite methods for multi-dimensional stochastic processes, Ph.D. thesis, U.B.C.

    Google Scholar 

  • M. Reimers (1987). Hyperfinite methods applied to the critical branching diffusion, Probab. Th. Rel. Fields 81, 11–27.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Reimers (1989). One dimenional stochastic partial differential equations and the branching measure diffusion, Probab. Th. Rel. Fields 81, 319–340.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Reimers (1992) A new result on the support of the Fleming-Viot process proved by non-standard construction, preprint.

    Google Scholar 

  • P. Ressel and W. Schmidtechen (1991). A new characterization of Laplace functionals and probability generating functionals, Prob. Th. Rel. Fields 88, 195–213.

    Article  MathSciNet  MATH  Google Scholar 

  • D. Revuz and M. Yor (1991). Continuous Martingales and Brownian Motion, Springer-Verlag.

    Google Scholar 

  • S. Roelly-Coppoletta (1986). A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastics 17, 43–65.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Roelly and S. Méléard (1990) Interacting branching measure processes, Proceedings: Stochastic Partial Differential Equations and Applications III, Trento, Italy, Springer-Verlag.

    Google Scholar 

  • S. Roelly-Coppoletta and A. Rouault (1989). Processus de Dawson-Watanabe conditioné par le futur lointain, C.R. Acad. Sci. Paris 309, 867–872.

    MathSciNet  MATH  Google Scholar 

  • S. Roelly and A. Rouault (1990). Construction et propriétés de martingales des branchements spatiaux interactifs, Int. Stat. Rev. 58, 173–189.

    Article  MATH  Google Scholar 

  • C.A. Rogers (1970). Hausdorff measures, Cambridge Univ. Press.

    Google Scholar 

  • L.C.G. Rogers and D. Williams (1987). Diffusions, Markov processes and Martingales, Vol. 2, Itp Calculus, Wiley.

    Google Scholar 

  • J. Rosen (1990). Renormalization and limit theorems for self-intersections of super-processes, preprint.

    Google Scholar 

  • B.L. Rozovskii (1990) Stochastic Evolution Equations, D. Reidel.

    Google Scholar 

  • S.M. Sagitov (1990). Multi-dimensional critical branching processes generated by large numbers of identical particles, Th. Prob. Appl. 35.

    Google Scholar 

  • K. Sato (1976a) Diffusion processes and a class of Markov chains related to population genetics, Osaka J. Math. 13, 631–659.

    MathSciNet  MATH  Google Scholar 

  • K. Sato (1976b). A class of Markov chains related to selection in population genetics, J. Math. Soc. Japan 28, 621–636.

    Article  MathSciNet  MATH  Google Scholar 

  • K. Sato (1978) Convergence to a diffusion of a multi-allelic model in population genetics, Adv. Appl. Prob. 10, 538–562.

    Article  MathSciNet  MATH  Google Scholar 

  • K.I. Sato (1983). Limit diffusion of some stepping stone models, J. Appl. Prob. 20, 460–471.

    Article  MATH  Google Scholar 

  • S. Sawyer (1976). Results for the stepping stone model for migration in population genetics, Ann. Prob. 4, 699–728.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Sawyer (1979). A limit theorem for patch size in a selectively neutral migration model, J. Appl. Prob. 16, 482–495.

    Article  MathSciNet  MATH  Google Scholar 

  • M.J. Sharpe (1988). General theory of Markov processes, Academic Press.

    Google Scholar 

  • B. Schmuland (1991). A result on the infinitely many neutral alleles diffusion model, J. Appl. Prob.

    Google Scholar 

  • T. Shiga (1980) An interacting system in population genetics, J. Math. Kyoto Univ. 20, 213–242.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga (1981) Diffusion processes in population genetics, J. Math. Kyoto Univ. 21, 133–151.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga (1982) Wandering phenomena in infinite allelic diffusion models, Adv. Appl. Prob. 14, 457–483.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Shiga (1982), Continuous time multi-allelic stepping stone models in population genetics, J. Math. Kyoto Univ. 22, 1–40.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga (1985) Mathematical results on the stepping stone model in population genetics, in Population Genetics and Molecular evolution, T. Ohta and K. Aoki, eds., Springer-Verlag.

    Google Scholar 

  • T. Shiga (1987a). Existence and uniqueness of solutions for a class of non-linear diffusion equations, J. Math. Kyoto Univ. 27-2, 195–215.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga (1987b). A certain class of infinite dimensional diffusion processes arising in population genetics, J. Math. Soc. Japan 30, 17–25.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Shiga (1988) Stepping stone models in population genetics and population dynamics, in S. Albeverio et al (eds.) Stochastic Processes in Physics and Engineering, 345–355.

    Google Scholar 

  • T. Shiga (1990a) A stochastic equation based on a Poisson system for a class of measure-valued diffusions, J. Math. Kyoto Univ. 30(1990), 245–279.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga (1990b) Two contrastive properties of solutions for one-dimensional stochastic partial differential equations, preprint.

    Google Scholar 

  • T. Shiga and A. Shimizu (1980) Infinite dimensional stochastic differential equations and their applications, J. Math. Kyoto Univ. 20, 395–416.

    MathSciNet  MATH  Google Scholar 

  • T. Shiga and K. Uchiyama (1986). Stationary states and the stability of the stepping stone model involving mutation and selection, Prob. Th. Rel. Fields 73, 87–117.

    Article  MathSciNet  MATH  Google Scholar 

  • N. Shimakura (1985). Existence and uniqueness of solutions for a diffusion model of intergroup selection, J. Math. Kyoto Univ. 25, 775–788.

    MathSciNet  MATH  Google Scholar 

  • A. Shimizu (1985). Diffusion approximation of an infinite allele model incorporating gene conversion, in Population genetics and molecular evolution, eds. T. Ohta and K. Aoki. Japan Sci. Soc. Press and Springer-Verlag.

    Google Scholar 

  • A. Shimizu (1987). Stationary distribution of a diffusion process taking values in probability distributions on the partitions, Lecture Notes in Biomath. 70, 100–114.

    Article  MathSciNet  MATH  Google Scholar 

  • A. Shimizu (1990). A measure valued diffusion process describing an n locus model incorporating gene conversion, Nagoya Math. J. 119, 81–92.

    MathSciNet  MATH  Google Scholar 

  • A.N. Shiryayev (1984). Probability, Springer-Verlag.

    Google Scholar 

  • M.L. Silverstein (1969). Continuous state branching semigroups, Z. Wahr. verw. Geb. 14, 96–112.

    Article  MathSciNet  MATH  Google Scholar 

  • D.W. Stroock and S.R.S. Varadhan (1979). Multidimensional diffusion processes, Springer-Verlag.

    Google Scholar 

  • S. Sugitani (1987). Some properties for the measure-valued branching diffusion processes, J. Math. Soc. Japan 41, 437–462.

    Article  MathSciNet  MATH  Google Scholar 

  • A.S. Sznitman (1991). Topics in Propagation of Chaos, Ecole d'été de Probabilités de Saint Flour, L.N.M. 1464, 165–251.

    MathSciNet  Google Scholar 

  • S.J. Taylor (1966). Multiple points for the sample paths of the symmetric stable process, Z. Wahr. verw. Geb. 5, 247–258.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Tavaré (1984). Line of descent and genealogical processes, and their applications in population genetics models, Theor. Pop. Biol. 26, 119–164.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Tavaré (1989). The genealogy of the birth, death and immigration process, in Mathematical Evolutionary Theory, ed. M.W. Feldman, 41–56.

    Google Scholar 

  • R. Tribe (1989). Path properties of superprocesses, Ph.D. thesis, U.B.C.

    Google Scholar 

  • R. Tribe (1991). The connected components of the closed support of super Brownian motion, Probab. Th. Rel. Fields 89, 75–87.

    Article  MathSciNet  MATH  Google Scholar 

  • R. Tribe (1992). The behavior of superprocesses near extinction, Ann. Probab. 20, 286–311.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Vaillancourt (1987). Interacting Fleming-Viot processes and related measure-valued processes, Ph.D. thesis, Carleton University.

    Google Scholar 

  • J. Vaillancourt (1988). On the existence of random McKean-Vlasov limits for triangular arrays of exchangeable diffusions, Stoch. Anal.

    Google Scholar 

  • J. Vaillancourt (1990a). Interacting Fleming-Viot processes, Stoch. Proc. Appl. 36, 45–57.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Vaillancourt (1990b). On the scaling theorem for interacting Fleming-Viot processes, Stoch. Proc. Appl. 36, 263–267.

    Article  MathSciNet  MATH  Google Scholar 

  • S.R.S. Varadhan (1984). Large Deviations and Applications, CBMS-NSF Regional Conf. 46, SIAM.

    Google Scholar 

  • A.D. Venttsel' (1985). Infinitesimal characteristics of Markov processes in a function space which describes the past, Th. Prob. Appl. 30, 661–676.

    Article  MathSciNet  MATH  Google Scholar 

  • A.D. Vent-tsel (1989). Refinement of the functional central limit theorem for stationary processes, Th. Prob. Appl. 34, 402–415.

    Article  MathSciNet  Google Scholar 

  • L. Véron (1981). Singular solutions of some nonlinear elliptic equations, Nonlinear Anal. Theory, Math. Appl. 5, 225–242.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Viot (1976). Solutions faibles d'equations aux dwrivwes partielles non lineaires, Thèse, Univ. Pierre et Marie Curie, Paris.

    Google Scholar 

  • J.B. Walsh (1986). An introduction to stochastic partial differential equations, in P.L. Hennequin (ed.), Ecole d'été de Probabilités de Saint-Flour XIV-1984, L.N.M. 1180, 265–439.

    Google Scholar 

  • F.S. Wang (1982a). Diffusion approximations of age-and-position dependent branching processes, Stoch. Proc. Appl. 13, 59–74.

    Article  MathSciNet  MATH  Google Scholar 

  • F.S. Wang (1982b). Probabilities of extinction of multiplicative measure diffusion processes with absorbing boundary, Indiana Univ. Math J. 31, 97–107.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Watanabe (1988). Averaging and fluctuations for parabolic equations with rapidly oscillating random coefficients, Probab. Th. Rel. Fields 77, 359–378.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Watanabe (1989). On the convergence of partial differential equations of parabolic type with rapidly oscillating coefficients, Appl. Math. Optim. 20, 81–96.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Watanabe (1968). A limit theorem of branching processes and continuous state branching, J. Math. Kyoto Univ. 8, 141–167.

    MathSciNet  MATH  Google Scholar 

  • S. Watanabe (1969). On two dimensional Markov processes with branching property, Trans. Amer. Math. Soc. 136, 447–466.

    Article  MathSciNet  MATH  Google Scholar 

  • G.A. Watterson (1976a) Reversibility and the age of an allele I. Moran's infinitely many neutral alleles model, Theor. Pop. Biol. 10, 239–253.

    Article  MathSciNet  MATH  Google Scholar 

  • G.A. Watterson (1976b). The stationary distribution of the infinitely many neutral alleles model, J. Appl. Prob. 13, 639–651.

    Article  MathSciNet  MATH  Google Scholar 

  • G.A. Watterson (1984) Lines of descent and the coaiescent, Theor. Pop. Biol. 10, 239–253.

    Article  MathSciNet  Google Scholar 

  • A.D. Wentzell (1992). On differentiability of the expectation of functionals of a Markov process, Stochastics and Stochastic Reports 39, 53–65.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Wright (1943). Isolation by distance, Genetics 28, 114–138.

    Google Scholar 

  • S. Wright (1949) Adaptation and selection. In Genetics, Paleontology and Evolution, ed. G.L. Jepson et al, 365–389, Princeton Univ. Press.

    Google Scholar 

  • Y. Wu (1991). Asymptotic behavior of two level branching processes, LRSP Tech. Report 179, Carleton Univ.

    Google Scholar 

  • Y. Wu (1991). Multilevel birth and death particle system and its continuous diffusion, LRSP Tech. Report 186, Carleton Univ.

    Google Scholar 

  • Y. Wu (1992). Dynamic particle systems and multilevel measure branching processes. Ph.D. thesis, Carleton University.

    Google Scholar 

  • T. Yamada and S. Watanabe (1971). On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto Univ. 11, 155–167, 553–563.

    MathSciNet  MATH  Google Scholar 

  • M. Yor (1974). Existence et unicité de diffusions à valeurs dans un espace de Hilbert, Ann. Inst. Henri Poincaré 10, 55–88.

    MATH  Google Scholar 

  • U. Zähle (1988a). Self-similar random measures I. Notion, carrying Hausdorff dimension and hyperbolic distribution, Probab. Th. Rel. Fields 80, 79–100.

    Article  MathSciNet  MATH  Google Scholar 

  • U. Zähle (1988b). The fractal character of localizable measure-valued processes I-random measures on product spaces, Math. Nachr. 136, 149–155.

    Article  MathSciNet  MATH  Google Scholar 

  • U. Zähle (1988c). The fractal character of localizable measure-valued processes II, Localizable processes and backward trees, Math. Nachr. 137, 35–48.

    Article  MathSciNet  MATH  Google Scholar 

  • U. Zähle (1988d). The fractal character of localizable measure-valued processes III. Fractal carrying sets of branching diffusions, Math. Nachr. 138, 293–311.

    Article  MathSciNet  MATH  Google Scholar 

  • H. Zessin (1983). The method of moments for random measures, Z. Wahr. verw. Geb. 62, 395–409.

    Article  MathSciNet  MATH  Google Scholar 

  • V.M. Zolotarev (1957). More exact statements of several theorems in the theory of branching processes, Th. Prob. Appl. 2, 245–253.

    Article  MathSciNet  MATH  Google Scholar 

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Paul-Louis Hennequin

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Dawson, D. (1993). Measure-valued Markov processes. In: Hennequin, PL. (eds) Ecole d'Eté de Probabilités de Saint-Flour XXI - 1991. Lecture Notes in Mathematics, vol 1541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084190

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  • DOI: https://doi.org/10.1007/BFb0084190

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