Branching Processes with Immigration and Related Topics

Abstract

This is a survey on the recent progresses in the study of branching processes with immigration, generalized Ornstein-Uhlenbeck processes, and affine Markov processes. We mainly focus on the applications of skew convolution semigroups and the connections in those processes.

Keywords

branching process / immigration / measure-valued process / affine process / Ornstein-Uhlenbeck process / skew convolution semigroup / stochastic equation / fluctuation limit / 60J80 / 60F05 / 60H20 / 60K37

References

[1] Athreya K. B., Ney P. E. Branching Processes, 1972, Berlin Heidelberg New York: Springer.

[2] Barros-Neto J., An Introduction to the Theory of Distributions, New York: Marcel Dekker, 1973 (Chinese translation, 1981)

[3] Bertoin J. Lévy Processes, 1996, Cambridge: Cambridge University Press.

[4] Bogachev V. I., Röckner M., Schmuland B. Generalized Mehler semigroups and applications. Probab. Theory Relat. Fields, 1996, 105: 193-225.

[5] Bojdecki T., Gorostiza L. G. Langevin equation for L′-valued Gaussian processes and fluctuation limits of infinite particle systems. Probab. Theory Relat. Fields, 1986, 73: 227-244.

[6] Chen M.-F. Ergodic convergence rates of Markov processes—eigenvalues, inequalities and ergodic theory. Proceedings of ICM 2002, Vol. III, 2002, Beijing: Higher Education Press, 25-40.

[7] Chen M.-F. From Markov Chains to Non-Equilibrium Particle Systems, 2004 2 Singapore: World Scientific.

[8] Chen M.-F. Eigenvalues, Inequalities and Ergodic Theory, 2004, London: Springer.

[9] Dawson D. A. Hannequin P. L. Measure-valued Markov Processes. Lecture Notes in Math., Vol. 1541, 1993, Berlin Heidelberg New York: Springer, 1-260.

[10] Dawson D. A., Fleischmann K. Critical branching in a highly fluctuating random medium. Probab. Theory Relat. Fields, 1991, 90: 241-274.

[11] Dawson D. A., Fleischmann K. A continuous super-Brownian motion in a super-Brownian medium. J. Theor. Probab., 1997, 10: 213-276.

[12] Dawson D. A., Fleischmann K., Gorostiza L. G. Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium. Ann. Probab., 1989, 17(3): 1083-1117.

[13] Dawson D. A., Li Z. H. Construction of immigration superprocesses with dependent spatial motion from one-dimensional excursions. Probab. Theory Relat. Fields, 2003, 127: 37-61.

[14] Dawson D. A., Li Z. H. Non-differentiable skew convolution semigroups and related Ornstein-Uhlenbeck processes. Potential Anal., 2004, 20: 285-302.

[15] Dawson D. A. and Li Z. H., Skew convolution semigroups and affine Markov processes, Ann. Probab., 2006 (in press)

[16] Dawson D. A., Li Z. H., Schmuland B., Sun W. Generalized Mehler semigroups and catalytic branching processes with immigration. Potential Anal., 2004, 21: 75-97.

[17] Dawson D. A., Li Z. H., Wang H. Superprocesses with dependent spatial motion and general branching densities. Electron. J. Probab., 2001, 6(25): 1-33.

[18] Dellacherie C., Meyer P. A. Probabilities and Potential, Chap. V–VIII, 1982, Amsterdam: North-Holland.

[19] Duffie D., Filipović D., Schachermayer W. Affine processes and applications in finance. Ann. Appl. Probab., 2003, 13: 984-1053.

[20] Dynkin E. B. Diffusions, Superdiffusions and Partial Differential Equations, 2002, Providence, RI: Amer. Math. Soc.

[21] Etheridge A. M. An Introduction to Superprocesses, 2000, Providence, RI: Amer. Math. Soc.

[22] Fitzsimmons P. J. Construction and regularity of measure-valued Markov branching processes. Isr. J. Math., 1988, 64: 337-361.

[23] Fitzsimmons P. J. On the martingale problem for measure-valued Markov branching processes. Seminar on Stochastic Processes, Vol. 1991, 1992, Boston, MA: Birkhäuser Boston, 39-51.

[24] Fu Z. F., Li Z. H. Measure-valued diffusions and stochastic equations with Poisson process. Osaka J. Math., 2004, 41: 727-744.

[25] Fuhrman M., Röckner M. Generalized Mehler semigroups: the non-Gaussian case. Potential Anal., 2000, 12: 1-47.

[26] Gorostiza L. G. and Li Z. H., Fluctuation limits of measure-valued immigration processes with small branching. In: Gonzalez-Burrios J. M. and Gorostiza L. G. (eds.), Aportaciones Matemáticas: Investigación, Vol. 14, Mexico: Sociedad Matemática Mexicana, 261–268

[27] Gorostiza L. G., Li Z. H. High density fluctuations of immigration branching particle systems. CMS Conference Proceedings, Ser. 26, 2000, Providence, RI: Amer. Math. Soc, 159-171.

[28] Grey D. R. Asymptotic behavior of continuous time, continuous state-space branching processes. J. Appl. Probab., 1974, 11: 669-677.

[29] Harris T. E. The Theory of Branching Processes, 1963, Berlin Heidelberg New York: Springer.

[30] Hong W. M. Long time behavior for the occupation time processes of a super-Brownian motion with random immigration. Stoch. Process. Appl., 2002, 102: 43-62.

[31] Hong W. M. Moderate deviation for the super-Brownian motion with super-Brownian immigration. J. Appl. Probab., 2002, 39: 829-838.

[32] Hong W. M. Large deviations for the super-Brownian motion with super-Brownian immigration. J. Theor. Probab., 2003, 16: 899-922.

[33] Hong W. M. Quenched mean limit theorems for the super-Brownian motion with super-Brownian immigration. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2005, 8: 383-396.

[34] Hong W. M., Li Z. H. A central limit theorem for super Brownian motion with super Brownian immigration. J. Appl. Probab., 1999, 36: 1218-1224.

[35] Hong W. M. and Li Z. H., Large and moderate deviations for occupation times of immigration superprocesses, 2005, 8: 593–603

[36] Ikeda N., Watanabe S. Stochastic Differential Equations and Diffusion Processes, 1989, Amsterdam: North-Holland.

[37] Jiřina M. Stochastic branching processes with continuous state space. Czechoslov. Math. J., 1958, 8: 292-313.

[38] Jiřina M., Branching processes with measure-valued states. In: Trans. Third Prague Conf. Information Theory, Statist. Decision Func., Random Process., Prague: Publ. House Czech. Acad. Sci., 1964, 333–357

[39] Kawazu K, Watanabe S Branching processes with immigration and related limit theorems. Theory Probab. Appl., 1971, 16: 36-54.

[40] Le Gall J-F Spatial Branching Processes, Random Snakes and Partial Differential Equations, 1999, Basel: Birkhäuser (Lectures in Mathematics ETH Zürich)

[41] Le Gall J.-F., Le Jan Y. Branching processes in Lévy processes: the exploration process. Ann. Probab., 1998, 26: 213-252.

[42] Li Z. H., Integral representations of continuous functions, Chinese Sci. Bull. (Chinese ed.), 1991, 36: 81–84, math.bnu.edu.cn/~lizh (English edn.: 1991, 36: 979–983)

[43] Li Z. H. Measure-valued branching processes with immigration. Stoch. Process. Appl., 1992, 43: 249-264.

[44] Li Z. H., Convolution semigroups associated with measure-valued branching processes, Chin. Sci. Bull. (Chinese edn.), 1995, 40: 2018–2021, math.bnu.edu.cn/~lizh (English edn.: 1996, 41: 276–280)

[45] Li Z. H. Immigration structures associated with Dawson-Watanabe superprocesses. Stoch. Process. Appl., 1996, 62: 73-86.

[46] Li Z. H. Immigration processes associated with branching particle systems. Adv. Appl. Probab., 1998, 30: 657-675.

[47] Li Z. H. Measure-valued immigration diffusions and generalized Ornstein-Uhlenbeck diffusions. Acta Math. Appl. Sin., 1999, 15: 310-320.

[48] Li Z. H. Asymptotic behavior of continuos time and state branching processes. J. Aust. Math. Soc., Ser. A, 2000, 68: 68-84.

[49] Li Z. H. Ornstein-Uhlenbeck type processes and branching processes with immigration. J. Appl. Probab., 2000, 37: 627-634.

[50] Li Z. H. Skew convolution semigroups and related immigration processes. Theory Probab. Appl., 2002, 46: 274-296.

[51] Li Z. H., A limit theorem of discrete Galton-Watson branching processes with immigration, J. Appl. Probab., 2006 (in press)

[52] Li Z. H., Shiga T. Measure-valued branching diffusions: immigrations, excursions and limit theorems. J. Math. Kyoto Univ., 1995, 35: 233-274.

[53] Li Z. H., Wang Z. K. Measure-valued branching process and immigration processes. Adv. Math. (China), 1999, 28: 105-134.

[54] Li Z. H. and Zhang M., Fluctuation limit theorems of immigration superprocesses with small branching, Stat. Probab. Lett., 2006 (in press)

[55] Linde W. Probability in Banach Spaces—Stable and Infinitely Divisible Distributions, 1986, New York: Wiley.

[56] van Neerven J. M. A. M. Continuity and representation of Gaussian Mehler semigroups. Potential Anal., 2000, 13: 199-211.

[57] Pakes A. G. Some limit theorems for continuous-state branching processes. J. Aust. Math. Soc., Ser. A, 1988, 44: 71-87.

[58] Pakes A. G. Revisiting conditional limit theorems for mortal simple branching processes. Bernoulli, 1999, 5: 969-998.

[59] Pitman J., Yor M. A decomposition of Bessel bridges. Z. Wahrscheinlichkeitstheor. Verw. Geb., 1982, 59: 425-457.

[60] Röckner M., Wang F. Y. Harnack and functional inequalities for generalized Mehler semigroups. J. Funct. Anal., 2003, 203: 237-261.

[61] Sato K. Lévy Processes and Infinitely Divisible Distributions, 1999, Cambridge: Cambridge University Press.

[62] Schmuland B., Sun W. On the equation μ_{t + s} = μ_s*T_sμ_t. Stat. Probab. Lett., 2001, 52: 183-188.

[63] Schaefer H. H. Topological Vector Spaces, 1980, Berlin Heidelberg New York: Springer.

[64] Sharpe M. J. General Theory of Markov Processes, 1988, New York: Academic Press.

[65] Shiga T., Watanabe S. Bessel diffusions as a one-parameter family of diffusion processes. Z. Wahrscheinlichkeitstheor. Verw. Geb., 1973, 27: 37-46.

[66] Walsh J. B. An introduction to stochastic partial differential equations. Lecture Notes in Math., Vol. 1180, 1986, Berlin Heidelberg New York: Springer, 226-439.

[67] Wang F. Y. Functional Inequalities, Markov Processes, and Spectral Theory, 2005, Beijing: Science Press.

[68] Wang F. Y., The stochastic order and critical phenomena for surperprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 2005 (in press)

[69] Wang F. Y., Dimension-free Harnack inequalities and applications, Front. Math. China, 2006, 1

[70] Wang H. State classification for a class of measure-valued branching diffusions in a Brownian medium. Probab. Theory Relat. Fields, 1997, 109: 39-55.

[71] Watanabe S., A limit theorem of branching processes and continuous state branching processes, J. Math. Kyoto Univ., 1968, 141–167

[72] Zhang M. Large deviation for super-Brownian motion with immigration. J. Appl. Probab., 2004, 41: 187-201.

[73] Zhang M. Moderate deviations for super-Brownian motion with immigration. Sci. China, Ser. A, 2004, 41: 440-452.

[74] Zhang M. Moderate deviation principles for the occupation time process of a super-Brownian motion with immigration. Chin. J. Contemp. Math., 2005, 26: 61-70.

[75] Zhang M., On the large deviation for Brownian branching particle system, J. Appl. Probab., 2005 (in press) (with the announcement to appear in Front. Math. China, 2006, 1)