Degenerate stochastic differential equations for catalytic branching networks

Sandra Kliem

Annales de l'I.H.P. Probabilités et statistiques (2009)

  • Volume: 45, Issue: 4, page 943-980
  • ISSN: 0246-0203

Abstract

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Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math.50 (2006) 323–383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.

How to cite

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Kliem, Sandra. "Degenerate stochastic differential equations for catalytic branching networks." Annales de l'I.H.P. Probabilités et statistiques 45.4 (2009): 943-980. <http://eudml.org/doc/78063>.

@article{Kliem2009,
abstract = {Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math.50 (2006) 323–383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.},
author = {Kliem, Sandra},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {stochastic differential equations; martingale problem; degenerate operators; catalytic branching networks; diffusions; semigroups; weighted Hölder norms; perturbations},
language = {eng},
number = {4},
pages = {943-980},
publisher = {Gauthier-Villars},
title = {Degenerate stochastic differential equations for catalytic branching networks},
url = {http://eudml.org/doc/78063},
volume = {45},
year = {2009},
}

TY - JOUR
AU - Kliem, Sandra
TI - Degenerate stochastic differential equations for catalytic branching networks
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2009
PB - Gauthier-Villars
VL - 45
IS - 4
SP - 943
EP - 980
AB - Uniqueness of the martingale problem corresponding to a degenerate SDE which models catalytic branching networks is proven. This work is an extension of the paper by Dawson and Perkins [Illinois J. Math.50 (2006) 323–383] to arbitrary catalytic branching networks. As part of the proof estimates on the corresponding semigroup are found in terms of weighted Hölder norms for arbitrary networks, which are proven to be equivalent to the semigroup norm for this generalized setting.
LA - eng
KW - stochastic differential equations; martingale problem; degenerate operators; catalytic branching networks; diffusions; semigroups; weighted Hölder norms; perturbations
UR - http://eudml.org/doc/78063
ER -

References

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  1. [1] S. R. Athreya, M. T. Barlow, R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations and super-Markov chains. Probab. Theory Related Fields 123 (2002) 484–520. Zbl1007.60053MR1921011
  2. [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. 357 (2005) 5001–5029 (electronic). Zbl1131.35008MR2165395
  3. [3] R. F. Bass. Diffusions and Elliptic Operators. Springer, New York, 1998. Zbl0914.60009MR1483890
  4. [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 (electronic). Zbl1007.60055MR1928092
  5. [5] R. F. Bass and E. A. Perkins. Degenerate stochastic differential equations arising from catalytic branching networks. Electron. J. Probab. 13 (2008) 1808–1885. Zbl1191.60070MR2448130
  6. [6] D. A. Dawson, A. Greven, F. den Hollander, R. Sun and J. M. Swart. The renormalization transformation for two-type branching models. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 1038–1077. Zbl1181.60122MR2469334
  7. [7] D. A. Dawson and E. A. Perkins. On the uniqueness problem for catalytic branching networks and other singular diffusions. Illinois J. Math. 50 (2006) 323–383 (electronic). Zbl1107.60045MR2247832
  8. [8] M. Eigen and P. Schuster. The Hypercycle: A Principle of Natural Self-organization. Springer, Berlin, 1979. 
  9. [9] J. Hofbauer and K. Sigmund. The Theory of Evolution and Dynamical Systems. London Math. Soc. Stud. Texts 7. Cambridge Univ. Press, Cambridge, 1988. Zbl0678.92010MR1071180
  10. [10] L. Mytnik. Uniqueness for a mutually catalytic branching model. Probab. Theory Related Fields 112 (1998) 245–253. Zbl0912.60076MR1653845
  11. [11] E. A. Perkins. Dawson–Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 125–324. Lecture Notes in Math. 1781. Springer, Berlin, 2002. Zbl1020.60075MR1915445
  12. [12] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 2. Reprint of the 2nd (1994) edition. Cambridge Univ. Press, Cambridge, 2000. Zbl0949.60003MR1780932
  13. [13] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. Grundlehren Math. Wiss. 233. Springer, Berlin, 1979. Zbl0426.60069MR532498

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