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 Electronic Journal of Probability > Vol. 9 (2004) > Paper 22 open journal systems 


Countable Systems of Degenerate Stochastic Differential Equations with Applications to Super-Markov Chains

Richard F Bass, University of Connecticut, USA
Edwin A. Perkins, The University of British Columbia


Abstract
We prove well-posedness of the martingale problem for an infinite-dimensional degenerate elliptic operator under appropriate Hölder continuity conditions on the coefficients. These martingale problems include large population limits of branching particle systems on a countable state space in which the particle dynamics and branching rates may depend on the entire population in a Hölder fashion. This extends an approach originally used by the authors in finite dimensions.


Full text: PDF

Pages: 634-673

Published on: October 6, 2004
Bibliography
  1. S.R. Athreya, M.T. Barlow, R.F. Bass, and E.A. Perkins, Degenerate stochastic differential equations and super-Markov chains. Probab. Th. Rel. Fields, 123 (2002) 484-520. MR1921011 (2003g:60096)

  2. S.R. Athreya R.F. Bass, and E.A. Perkins, Hölder norm estimates for elliptic operators on finite and infinite dimensional spaces, Trans. Amer. Math. Soc., to appear. Math Review not yet available.
  3. R.F. Bass, Diffusions and Elliptic Operators. New York, Springer-Verlag, 1997. MR1483890 (99h:60136)

  4. R.F. Bass and E.A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains. Trans. Amer. Math. Soc., 355 (2003) 373-405. MR1928092 (2003m:60144)

  5. P. Cannarsa and G. Da Prato, Infinite-dimensional elliptic equations with Hölder-continuous coefficients. Adv. Differential Equations, 1 (1996) 425-452. MR1401401 (97g:35174)

  6. G. Da Prato, Some results on elliptic and parabolic equations in Hilbert spaces. Rend. Mat. Acc. Lincei, 7 (1996) 181-199. MR1454413 (98g:35206)

  7. G. Da Prato and J. Zabczyk, Second order partial differential equations in Hilbert spaces. Cambridge University Press, Cambridge, 2002. MR1985790 (2004e:47058)

  8. D. A. Dawson and P. March, Resolvent estimates for Fleming-Viot operators, and uniqueness of solutions to related martingale problems. J. Funct. Anal., 132 (1995) 417-472. MR1347357 (97a:60105)

  9. S.N. Ethier and T.G. Kurtz, Markov Processes. Characterization and Convergence. Wiley, New York, 1986. MR0838085 (88a:60130)

  10. A. Lunardi, An interpolation method to characterize domains of generators of semigroups. Semigroup Forum, 53 (1996) 321-329. MR1406778 (98a:47040)

  11. S. Méléard and S. Roelly, Interacting branching measure processes. Stochastic Partial Differential Equations and Applications, 246-256, Longman, Harlow, 1992. MR1222702 (94g:60159)

  12. E. Perkins, Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125-324, Lecture Notes in Math., 1781, Springer, Berlin, 2002. MR1915445 (2003k:60104)

  13. T. Shiga and A. Shimizu, Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ., 20 (1980) 395-416. MR0591802 (82i:60110)

  14. D.W. Stroock and S.R.S.Varadhan, Multidimensional Diffusion Processes. New York, Springer-Verlag, 1979. MR0532498 (81f:60108)

  15. L. Zambotti, An analytic approach to existence and uniqueness for martingale problems in infinite dimensions. Probab. Theory Related Fields, 118 (2000) 147-168. MR1790079 (2001h:60116)













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Electronic Journal of Probability. ISSN: 1083-6489