Electronic Journal of Probability

Degenerate stochastic differential equations arising from catalytic branching networks

Richard Bass and Edwin Perkins

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Abstract

We establish existence and uniqueness for the martingale problem associated with a system of degenerate SDE's representing a catalytic branching network. The drift and branching coefficients are only assumed to be continuous and satisfy some natural non-degeneracy conditions. We assume at most one catalyst per site as is the case for the hypercyclic equation. Here the two-dimensional case with affine drift is required in work of [DGHSS] on mean fields limits of block averages for 2-type branching models on a hierarchical group. The proofs make use of some new methods, including Cotlar's lemma to establish asymptotic orthogonality of the derivatives of an associated semigroup at different times, and a refined integration by parts technique from [DP1].

Article information

Source
Electron. J. Probab., Volume 13 (2008), paper no. 60, 1808-1885.

Dates
Accepted: 4 October 2008
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819133

Digital Object Identifier
doi:10.1214/EJP.v13-568

Mathematical Reviews number (MathSciNet)
MR2448130

Zentralblatt MATH identifier
1191.60070

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05]
Secondary: 35R15: Partial differential equations on infinite-dimensional (e.g. function) spaces (= PDE in infinitely many variables) [See also 46Gxx, 58D25] 60H30: Applications of stochastic analysis (to PDE, etc.)

Keywords
stochastic differential equations perturbations resolvents Cotlar's lemma catalytic branching martingale problem degenerate diffusions

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bass, Richard; Perkins, Edwin. Degenerate stochastic differential equations arising from catalytic branching networks. Electron. J. Probab. 13 (2008), paper no. 60, 1808--1885. doi:10.1214/EJP.v13-568. https://projecteuclid.org/euclid.ejp/1464819133


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References

  • S.R. Athreya, M.T. Barlow, R.F. Bass, and E.A. Perkins, Degenerate stochastic differential equations and super-Markov chains. Prob. Th. Rel. Fields 123 (2002), 484–520.
  • R.F. Bass, Probabilistic Techniques in Analysis, Springer, Berlin 1995.
  • R.F. Bass, Diffusions and Elliptic Operators, Springer, Berlin, 1998.
  • 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.
  • D.A. Dawson and K. Fleischmann, Catalytic and mutually catalytic branching. In: Infinite Dimensional Stochastic analysis, Ned. Acak. Wet., Vol. 52, R. Neth. Acad. Arts Sci., Amsterdam, 2000, pp. 145–170.
  • D.A. Dawson, K. Fleischmann, and J. Xiong, Strong uniqueness for cyclically catalytic symbiotic branching diffusions. Statist. Probab. Lett. 73 (2005) 251–257.
  • D.A. Dawson, A. Greven, F. den Hollander, Rongfeng Sun, and J.M. Swart, The renormalization transformation for two-type branching models, to appear Ann. de l'Inst. H. Poincaré, Prob. et Stat.
  • 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.
  • D.A. Dawson and E. A. Perkins, Long-time behaviour and coexistence in a mutually catalytic branching model. Ann. Probab. 26 (1998) 1088–1138.
  • M. Eigen and P. Schuster, The Hypercycle: a Principle of Natural Self-organization, Springer, Berlin, 1979.
  • C. Fefferman, Recent progress in classical Fourier analysis. Proceedings of the International Congress of Mathematicians, Vol. 1, pp. 95–118. Montréal, Canadian Math. Congress, 1975. (58#23308)
  • K. Fleischmann and J. Xiong, A cyclically catalytic super-Brownian motion. Ann. Probab. 29 (2001) 820–861.
  • S. Kliem, Degenerate stochastic differential equations for catalytic branching networks, to appear in Ann. de l'Inst. H. Poincaré, Prob. et Stat.
  • L. Mytnik, Uniqueness for a mutually catalytic branching model. Prob. Th. Rel. Fields 112 (1998) 245-253.
  • E.A. Perkins, Dawson-Watanabe Superprocesses and Measure-Valued Diffusions, In: Lectures on Probability and Statistics, Ecole d”Eté de Probabilités de Saint-Flour XXIX (1999), LNM vol. 1781, Springer-Verlag, Berlin, 2002, pp. 125–324.
  • D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Berlin, Springer-Verlag, 1991.
  • D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, Berlin 1979.
  • A. Torchinsky, Real-variable methods in harmonic analysis. Academic Press, Orlando, FL, 1986.