Recurrence properties of Lotka-Volterra models with random fluctuations

H. KESTEN; Y. OGURA

doi:10.2969/jmsj/03320335

H. KESTEN, Y. OGURA

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JOURNAL FREE ACCESS

1981 Volume 33 Issue 2 Pages 335-366

DOI https://doi.org/10.2969/jmsj/03320335

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Published: 1981 Received: July 16, 1979 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. M. Bhattacharya, Criteria for recurrence and existence of invariant measures of multi-dimensional diffusions, Ann. Probability, 6 (1978), 541-553 and 8 (1980), 1194-1195.
2) L. Breiman, Probability, Addison-Wesley Publ. Co., 1968.
3) D. L. Burkholder, Exit times of Brownian motion, harmonic majorization and Hardy spaces, Advances Math., 26 (1977), 182-205.
4) E. B. Dynkin, Markov processes, Springer Verlag, 1965.
5) A. Friedman, Stochastic differential equations and applications, Academic Press, 1976.
6) J. H. Gillespie and M. Turelli, Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology, Theoret. Population Biology, 17 (1980), 167-189.
7) J. Hutton, Recurrence and transience of two-dimensional linear birth and death processes, Ph. D. Thesis, Cornell University, 1979; see also Advances in Appl. Probability, (1980).
8) N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.
9) H. Kesten, Recurrence criteria for multi-dimensional Markov chains and multidimensional birth and death processes, Advances in Appl. Probability, 8 (1976), 58-87.
10) R. Z. Khasminskii, Stability of a system of differential equations with random perturbations of their parameters, Nauka, Moscow, 1969.
11) P. Mandl, Analytical treatment of one-dimensional Markov processes, Springer Verlag, 1968.
12) G. Maruyama and H. Tanaka, Ergodic property of n-dimensional recurrent Markov processes, Mem. Fac. Sci. Kyushu Univ. Ser. A, 13 (1959), 157-172.
13) H. P. McKean Jr., Stochastic integrals, Academic Press, 1969.
14) P. Polansky, Invariant distributions for multi-population models in random environments, preprint, 1978.
15) M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178.

Right : [1] R. M. Bhattacharya, Criteria for recurrence and existence of invariant measures of multi-dimensional diffusions, Ann. Probability, 6 (1978), 541-553 and 8 (1980), 1194-1195.
[2] L. Breiman, Probability, Addison-Wesley Publ. Co., 1968.
[3] D. L. Burkholder, Exit times of Brownian motion, harmonic majorization and Hardy spaces, Advances Math., 26 (1977), 182-205.
[4] E. B. Dynkin, Markov processes, Springer Verlag, 1965.
[5] A. Friedman, Stochastic differential equations and applications, Academic Press, 1976.
[6] J. H. Gillespie and M. Turelli, Conditions for the existence of stationary densities for some two-dimensional diffusion processes with applications in population biology, Theoret. Population Biology, 17 (1980), 167-189.
[7] J. Hutton, Recurrence and transience of two-dimensional linear birth and death processes, Ph. D. Thesis, Cornell University, 1979; see also Advances in Appl. Probability, (1980).
[8] N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.
[9] H. Kesten, Recurrence criteria for multi-dimensional Markov chains and multidimensional birth and death processes, Advances in Appl. Probability, 8 (1976), 58-87.
[10] R. Z. Khasminskii, Stability of a system of differential equations with random perturbations of their parameters, Nauka, Moscow, 1969.
[11] P. Mandl, Analytical treatment of one-dimensional Markov processes, Springer Verlag, 1968.
[12] G. Maruyama and H. Tanaka, Ergodic property of n-dimensional recurrent Markov processes, Mem. Fac. Sci. Kyushu Univ. Ser. A, 13 (1959), 157-172.
[13] H. P. McKean Jr., Stochastic integrals, Academic Press, 1969.
[14] P. Polansky, Invariant distributions for multi-population models in random environments, preprint, 1978.
[15] M. Turelli, Random environments and stochastic calculus, Theoret. Population Biology, 12 (1977), 140-178.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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