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Remarks on geometric properties of SQG sharp fronts and $\alpha$-patches
A transformation of Markov jump processes and applications in genetic study
1. | Academy of Math and Systems Science, CAS, Zhong-guan-cun East Road 55, Beijing 100190, China, China |
References:
[1] |
F. Ball and G. F. Yeo, Lumpability and marginalisability for continuous-time Markov chains, J. Appl. Probab., 30 (1993), 518-528.
doi: 10.2307/3214762. |
[2] |
R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968. |
[3] |
C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist., 29 (1958), 1112-1122.
doi: 10.1214/aoms/1177706444. |
[4] |
A. Y. Chen, P. Pollett, H. J. Zhang and B. Cairns, Uniqueness criteria for continuous-time Markov chains with general transition structures, Adv. Appl. Prob., 37 (2005), 1056-1074.
doi: 10.1239/aap/1134587753. |
[5] |
M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004.
doi: 10.1142/9789812562456. |
[6] |
M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24. |
[7] |
X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination, Annals of Statistics, to appear. |
[8] | |
[9] |
E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965. |
[10] |
L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra Appl., 404 (2005), 85-117.
doi: 10.1016/j.laa.2005.02.007. |
[11] |
J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237.
doi: 10.1214/aoms/1177704261. |
[12] |
S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992. |
[13] |
Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159. |
[14] |
Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005. |
[15] |
O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002.
doi: 10.1007/978-1-4757-4015-8. |
[16] |
J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976. |
[17] |
J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998. |
[18] |
M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596. |
[19] |
M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988. |
[20] |
J. P. Tian and X. S. Lin, Colored coalescent theory, Discrete Contin. Dyn. Syst. suppl., (2005), 833-845.
doi: 10.1007/s11538-009-9428-4. |
[21] |
J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702.
doi: 10.1080/07362990600632045. |
[22] |
Y. Wang, Y. Zhou, L. F. Li, X. Chen, Y. T. Liu, Z. M. Ma and S. H. Xu, A new method for modeling coalescent processes with recombination, preprint. |
[23] |
Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978. |
show all references
References:
[1] |
F. Ball and G. F. Yeo, Lumpability and marginalisability for continuous-time Markov chains, J. Appl. Probab., 30 (1993), 518-528.
doi: 10.2307/3214762. |
[2] |
R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Academic Press, New York-London, 1968. |
[3] |
C. J. Burke and M. Rosenblatt, A Markovian function of a Markov chain, Ann. Math. Statist., 29 (1958), 1112-1122.
doi: 10.1214/aoms/1177706444. |
[4] |
A. Y. Chen, P. Pollett, H. J. Zhang and B. Cairns, Uniqueness criteria for continuous-time Markov chains with general transition structures, Adv. Appl. Prob., 37 (2005), 1056-1074.
doi: 10.1239/aap/1134587753. |
[5] |
M. F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, 2nd edition, World Scientific, Singapore, 2004.
doi: 10.1142/9789812562456. |
[6] |
M. F. Chen and X. G. Zheng, Uniquness criterion for q-processes, Sci. Sin., 26 (1983), 11-24. |
[7] |
X. Chen, Z. M. Ma and Y. Wang, Markov jump processes in modeling coalsecent with recombination, Annals of Statistics, to appear. |
[8] | |
[9] |
E. B. Dynkin, Markov processes, Springer, Berlin Heidelberg, 1965. |
[10] |
L. Gurvits and J. Ledoux, Markov property for a function of a Markov chain: A linear algebra approach, Linear Algebra Appl., 404 (2005), 85-117.
doi: 10.1016/j.laa.2005.02.007. |
[11] |
J. Hachigian, Collapsed Markov chains and the Chapman-Kolmogorov equation, Ann. Math. Statist., 34 (1963), 233-237.
doi: 10.1214/aoms/1177704261. |
[12] |
S. W. He, J. G. Wang and J. A. Yan, Semimartingale Theory and Stochastic Calculus, Science Press, Beijing, 1992. |
[13] |
Z. T. Hou, The criterion for uniqueness of a Q process, Sci. Sinica, 17 (1974), 141-159. |
[14] |
Z. T. Hou and G. X. Liu, Markov Skeleton Processes and Their Applications, Science Press, Beijing, 2005. |
[15] |
O. Kallenberg, Foundations of Modern Probability, Springer, New York, 2002.
doi: 10.1007/978-1-4757-4015-8. |
[16] |
J. G. Kemeny and J. L. Snell, Finite Markov Chains, Springer, New York, 1976. |
[17] |
J. R. Norris, Markov Chains, Cambridge University Press, Cambridge, 1998. |
[18] |
M. Rosenblatt, Functions of a Markov process that are Markovian, J. Math. Mech., 8 (1959), 585-596. |
[19] |
M. Sharpe, General Theory of Markov Processes, Academic Press, Inc., Boston, MA, 1988. |
[20] |
J. P. Tian and X. S. Lin, Colored coalescent theory, Discrete Contin. Dyn. Syst. suppl., (2005), 833-845.
doi: 10.1007/s11538-009-9428-4. |
[21] |
J. P. Tian and D. Kanna, Lumpability and commutativity of Markov processes, Stoch. Anal. Appl., 24 (2006), 685-702.
doi: 10.1080/07362990600632045. |
[22] |
Y. Wang, Y. Zhou, L. F. Li, X. Chen, Y. T. Liu, Z. M. Ma and S. H. Xu, A new method for modeling coalescent processes with recombination, preprint. |
[23] |
Z. K. Wang, The Theory of Stochastic Processes, (Chinese) Science Press, Beijing, 1978. |
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