The Annals of Probability

Recurrence and transience of branching diffusion processes on Riemannian manifolds

Alexander Grigor'yan and Mark Kelbert

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Abstract

We relate the recurrence and transience of a branching diffusion process on a Riemannian manifold M to some properties of a linear elliptic operator onM (including spectral properties). There is a trade-off between the tendency of the transient Brownian motion to escape and the birth process of the new particles. If the latter has a high enough intensity then it may override the transience of the Brownian motion, leading to the recurrence of the branching process, and vice versa. In the case of a spherically symmetric manifold, the critical intensity of the population growth can be found explicitly.

Article information

Source
Ann. Probab. Volume 31, Number 1 (2003), 244-284.

Dates
First available in Project Euclid: 26 February 2003

Permanent link to this document
http://projecteuclid.org/euclid.aop/1046294311

Digital Object Identifier
doi:10.1214/aop/1046294311

Mathematical Reviews number (MathSciNet)
MR1959793

Zentralblatt MATH identifier
1014.60081

Subjects
Primary: 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60] 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords
Branching process Brownian motion Riemannian manifold recurrence transience maximum principle gauge

Citation

Grigor'yan, Alexander; Kelbert, Mark. Recurrence and transience of branching diffusion processes on Riemannian manifolds. Ann. Probab. 31 (2003), no. 1, 244--284. doi:10.1214/aop/1046294311. http://projecteuclid.org/euclid.aop/1046294311.


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References

  • [1] ATHREy A, K. and NEY, P. (1972). Branching Processes. Springer, New York.
  • [2] BERESTy CKI, H., NIRENBERG, L. and VARADHAN, S. R. S. (1994). The principal eigenvalue and maximum principle for second-order elliptic operators in general domains. Comm. Pure Appl. Math. 47 47-93.
  • [3] CHUNG, K. L. and ZHAO, Z. (1985). From Brownian Motion to Schrödinger's Equation. Springer, New York.
  • [4] Dy NKIN, E. B. (1991). Branching particle sy stems and superprocesses. Ann. Probab. 19 1157- 1194.
  • [5] GREENE, R. and WU, W. (1979). Function Theory of Manifolds which Possess a Pole. Lecture Notes in Math. 699. Springer, Berlin.
  • [6] GRIGOR'YAN, A. (1986). On stochastically complete manifolds. DAN SSSR 290 534-537 (in Russian). [English translation (1987) Soviet Math. Dokl. 34 310-313.]
  • [7] GRIGOR'YAN, A. (1991). The heat equation on non-compact Riemannian manifolds. Matem. Sbornik 182 55-87 (in Russian). [English translation (1992) Math. USSR Sb. 72 47-77.]
  • [8] GRIGOR'YAN, A. (1999). Analy tic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. 36 135-249.
  • [9] GRIGOR'YAN, A. and HANSEN, W. (1998). A Liouville property for Schrödinger operators. Math. Ann. 312 659-716.
  • [10] GRIGOR'YAN, A. and HANSEN, W. (1999). A Green function for a Schrödinger equation. Unpublished manuscript.
  • [11] GRIGOR'YAN, A. and TELCS, A. (2001). Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 452-510.
  • [12] HANSEN, W. Harnack inequalities for Schrödinger operators. Ann. Scuola Norm. Sup. Pisa 28 413-470.
  • [13] HARDY, G. H., LITTLEWOOD, J. E. and PÓLy A, G. (1934). Inequalities. Cambridge Univ. Press.
  • [14] KARPELEVICH, F. I., PECHERSKY, E. A. and SUHOV, YU. M. (1998). A phase transition for hy perbolic branching processes. Comm. Math. Phy s. 195 627-642.
  • [15] KARPELEVICH, F. I. and SUHOV, YU. M. (1997). Boundedness of one-dimensional branching Markov processes. J. Appl. Math. Stochastic Anal. 10 307-332.
  • [16] KAZDAN, J. (1982). Deformation to positive scalar curvature on complete manifolds. Math. Ann. 261 227-234.
  • [17] LALLEY, S. P. and SELLKE, T. (1997). Hy perbolic branching Brownian motion. Probab. Theory Related Fields 108 171-192.
  • [18] LI, P. and YAU, S.-T. (1986). On the parabolic kernel of the Schrödinger operator. Acta Math. 156 153-201.
  • [19] LITTMAN, N., STAMPACCIA, G. and WEINBERGER, H. F. (1963). Regular points for elliptic equations with discontinuous coefficients. Ann. Scuola Norm. Sup. Pisa 17 43-77.
  • [20] MACHADO, F. P., MENSHIKOV, M. V. and POPOV, S. YU. (2001). Recurrence and transience of multity pe branching random walks. Stochastic Process. Appl. 91 21-37.
  • [21] PINSKY, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Univ. Press.
  • [22] SULLIVAN, D. (1987). Related aspects of positivity in Riemannian geometry. J. Differential Geom. 25 327-351.
  • UNIVERSITY OF WALES, SWANSEA SINGLETON PARK SWANSEA SA2 8PP UNITED KINGDOM E-MAIL: m.kelbert@swansea.ac.uk