CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

doi:10.1007/s10473-021-0504-7

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (5): 1445-1473.doi: 10.1007/s10473-021-0504-7

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

陈舒凯, 李增沪

Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

收稿日期:2020-03-20 修回日期:2020-11-27 出版日期:2021-10-25 发布日期:2021-10-21
通讯作者: Shukai CHEN E-mail:skchen@mail.bnu.edu.cn
作者简介:Zenghu LI,E-mail:lizh@bnu.edu.cn
基金资助:
The authors were supported by the National Key R&D Program of China (2020YFA0712900) and the National Natural Science Foundation of China (11531001).

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

Shukai CHEN, Zenghu LI

Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received:2020-03-20 Revised:2020-11-27 Online:2021-10-25 Published:2021-10-21
Contact: Shukai CHEN E-mail:skchen@mail.bnu.edu.cn
Supported by:
The authors were supported by the National Key R&D Program of China (2020YFA0712900) and the National Natural Science Foundation of China (11531001).

摘要/Abstract

摘要： A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

关键词: mixed state branching process, weak convergence, stochastic equation system, Wasserstein-type distance, stationary distribution

Abstract: A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes. The process can also be obtained by the pathwise unique solution to a stochastic equation system. From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup. Meanwhile, we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

Key words: mixed state branching process, weak convergence, stochastic equation system, Wasserstein-type distance, stationary distribution

中图分类号:

60J80

陈舒凯, 李增沪. CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS[J]. 数学物理学报(英文版), 2021, 41(5): 1445-1473.

Shukai CHEN, Zenghu LI. CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS[J]. Acta mathematica scientia,Series B, 2021, 41(5): 1445-1473.

参考文献

[1] Athreya K B, Ney P E. Branching Processes. Berlin:Springer, 1972
[2] Bertoin J, Fontbona J, Martínez S. On prolific indivials in a supercritical continuous-state branching process. J Appl Probab, 2008, 45:714-726
[3] Bertoin J, Le Gall J F. Stochastic flows associated to coalescent processes III:Limit theorems. Illinois J Math, 2006, 50:147-181
[4] Chen M. From Markov Chains to Non-Equilibrium Particle Systems. 2nd ed. Singapore:World Sci, 2004
[5] Dawson D A, Li Z. Skew convolution semigroups and affine Markov processes. Ann Probab, 2006, 34:1103-1142
[6] Dawson D A, Li Z. Stochastic equations, flows and measure-valued processes. Ann Probab, 2012, 40:813-857
[7] Dellacherie C, Meyer P A. Probabilites and Potential. Chapters V-VIII. Amsterdam:NorthHolland, 1982
[8] Ethier S N, Kurtz T G. Markov Processes:Characterization and Convergence. New York:John Wiley and Sons Inc, 1986
[9] Feketa D, Fontbona J, Kyprianou A E. Skeletal stochastic differential equations for continuous-state branching processes. J Appl Probab, 2019, 56:1122-1150
[10] Feketa D, Fontbona J, Kyprianou A E. Skeletal stochastic differential equations for superprocesses. J Appl Probab, 2020, 57:1111-1134
[11] Feller W. Diffusion processes in genetics//Proceedings 2nd Berkeley Symp Math Statist Probab. Berkeley and Los Angeles:University of California Press, 1950:227-246
[12] Friesen M, Jin P, Kremer J, Rüdiger B. Exponential ergodicity for stochastic equations of nonnegative processes with jumps. 2019[2019-07-15]. https://arxiv.org/abs/1902.02833
[13] Fu Z, Li Z. Stochastic equations of non-negative processes with jumps. Stochastic Process Appl, 2010, 120:306-330
[14] He X, Li Z. Distributions of jumps in a continuous-state branching process with immigration. J Appl Probab, 2016, 53:1166-1177
[15] Ikeda N, Watanabe S. Stochastic Differential Equations and Diffusion Processes. Amsterdam/Tokyo:NorthHolland/Kodansha, 1989
[16] Jiřina M. Stochastic branching processes with continuous state space. Czechoslovak Math J, 1958, 8:292-313
[17] Ji L, Li Z. Moments of continuous-state branching processes with or without immigration. Acta Math Appl Sin Engl Ser, 2020, (2):361-373
[18] Ji L, Zheng X. Moments of continuous-state branching processes in Lévy random environments. Acta Math Sci, 2019, 39B(3):781-796
[19] Jiao Y, Ma C, Scotti S. Alpha-CIR model with branching processes in sovereign interest rate modeling. Finance Stoch, 2017, 21:789-813
[20] Jin P, Kremer J, Rüdiger B. Existence of limiting distribution for affine processes. J Math Anal Appl, 2020, 486:123912, 31 pp
[21] Kawazu K, Watanabe S. Branching processes with immigration and related limit theorems. Theory Probab Appl, 1971, 16:36-54
[22] Li Z. A limit theorem for discrete Galton-Watson branching processes with immigration. J Appl Probab, 2006, 43:289-295
[23] Li Z. Measure-Valued Branching Markov Processes. Berlin:Springer, 2011
[24] Li Z, Ma C. Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model. Stochastic Process Appl, 2015, 125:3196-3233
[25] Li Z. Continuous-state branching processes with immigration//Jiao Y. From Probability to Finance, Mathematical Lectures from Peking University. Singapore:Springer, 2020:1-69
[26] Li Z. Ergodicities and exponential ergodicities of Dawson-Watanabe type processes. 2020[2020-02-22]. https://arxiv.org/abs/2002.09111
[27] Ma C. A limit theorem of two-type Galton-Watson branching processes with immigration. Stat Prob Lett, 2009, 79:1710-1716
[28] Ma R. Stochastic equations for two-type continuous-state branching processes with immigration. Acta Math Sinica Engl Ser, 2013, 29:287-294
[29] Ma R. Stochastic equations for two-type continuous-state branching processes with immigration and competition. Stat Prob Lett, 2014, 91:83-89
[30] Pardoux E. Probabilistic Models of Population Evolution:Scaling Limits, Genealogies and Interactions. Switzerland:Springer, 2016
[31] Pinsky M A. Limit theorems for continuous state branching processes with immigration. Bull Amer Math Soc, 1972, 78:242-244
[32] Sato K, Yamazato M. Operator-self-decomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type. Stochastic Process Appl, 1984, 17:73-100

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 5

[1]	韩七星, 蒋达清. DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE[J]. 数学物理学报(英文版), 2020, 40(6): 1883-1896.
[2]	赵文强, 张一进. UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON-AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION[J]. 数学物理学报(英文版), 2020, 40(4): 921-933.
[3]	刘丽雅, 蒋达清, Tasawar HAYAT, Bashir AHMAD. DYNAMICS OF A HEPATITIS B MODEL WITH SATURATED INCIDENCE[J]. 数学物理学报(英文版), 2018, 38(6): 1731-1750.
[4]	付静, 蒋达清, 史宁中, Tasawar HAYAT, Ahmed ALSAEDI. QUALITATIVE ANALYSIS OF A STOCHASTIC RATIO-DEPENDENT HOLLING-TANNER SYSTEM[J]. 数学物理学报(英文版), 2018, 38(2): 429-440.
[5]	刘群, 蒋达清, 史宁中, Tasawar HAYAT, Ahmed ALSAEDI. DYNAMICAL BEHAVIOR OF A STOCHASTIC HBV INFECTION MODEL WITH LOGISTIC HEPATOCYTE GROWTH[J]. 数学物理学报(英文版), 2017, 37(4): 927-940.
[6]	Nermina MUJAKOVIC, Nelida CRNJARIC-ZIC. GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY[J]. 数学物理学报(英文版), 2016, 36(6): 1541-1576.
[7]	刘三洋, 贺慧敏, 陈汝栋. APPROXIMATING SOLUTION OF 0 ∈T(x) FOR AN H-MONOTONE OPERATOR IN HILBERT SPACES[J]. 数学物理学报(英文版), 2013, 33(5): 1347-1360.
[8]	蔡钢, 步尚全. WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES[J]. 数学物理学报(英文版), 2013, 33(1): 303-320.
[9]	甘师信, 陈平炎. SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF ARRAYS OF NOD RANDOM VARIABLES[J]. 数学物理学报(英文版), 2012, 32(6): 2388-2400.
[10]	戴洪帅, 李育强. STABLE SUB-GAUSSIAN MODELS CONSTRUCTED BY POISSON PROCESSES[J]. 数学物理学报(英文版), 2011, 31(5): 1945-1958.
[11]	叶俊. QUASI-STATIONARY DISTRIBUTIONS FOR THE RADIAL ORNSTEIN-UHLENBECK PROCESSES[J]. 数学物理学报(英文版), 2008, 28(3): 513-522.
[12]	孔繁超, 唐启鹤. NOTES ON ERDOS&rsquo\|CONJECTURE[J]. 数学物理学报(英文版), 2000, 20(4): 533-541.
[13]	韩东. THE NEAR-CRITICAL AND SUPER-CRITICAL ASYMPTOTIC BEHAVIOR IN THE THERMODYNAMIC LIMIT OF REVERSIBLE RANDOM POLYMERIZATION PROCESSES[J]. 数学物理学报(英文版), 2000, 20(3): 380-389.
[14]	王启华. SOME LARGE SAMPLE PROPERTIES OF AN ESTIMATOR OF THE HAZARD FUNCTION FROM RANDOMLY CENSORED DATA[J]. 数学物理学报(英文版), 1997, 17(2): 230-240.
[15]	韩东. THE STATIONARY DISTRIBUTION OF A CONTINUOUS-TIME RANDOM GRAPH PROCESS WITH INTERACTING EDGES[J]. 数学物理学报(英文版), 1994, 14(S1): 98-102.