The Annals of Probability

On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching

János Engländer and Ross G. Pinsky

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Abstract

In this paper, we construct a measure-valued diffusion on $D\subseteq \mathbb{R^d}$ whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator

[L = 1/2 \nabla \cdot a\nabla + b \cdot \nabla \text{ on } D \subseteq \mathbb{R}^d

and whose spatially dependent branching term is of the form $\beta(x)z-\alpha(x)z^2,x \inD$,where $\beta$ satisfies a very general condition and $\alpha> 0$. In the case that $\alpha$ and $\beta$ are bounded from above, we show that the measure-valued process can also be obtained as a limit of approximating branching particle systems.

We give criteria for extinction/survival, recurrence/transience of the support, compactness of the support, compactness of the range, and local extinction for the measure-valued diffusion. We also present a number of examples which reveal that the behavior of the measure-valued diffusion may be dramatically different from that of the approximating particle systems.

Article information

Source
Ann. Probab., Volume 27, Number 2 (1999), 684-730.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677383

Digital Object Identifier
doi:10.1214/aop/1022677383

Mathematical Reviews number (MathSciNet)
MR1698955

Zentralblatt MATH identifier
0979.60078

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60J60

Keywords
Measure-valued process superprocess diffusion process log-Laplace equation branching h-transform

Citation

Engländer, János; Pinsky, Ross G. On the Construction and Support Properties of Measure-Valued Diffusions on $D \subseteq \mathbb{R}^d$ with Spatially Dependent Branching. Ann. Probab. 27 (1999), no. 2, 684--730. doi:10.1214/aop/1022677383. https://projecteuclid.org/euclid.aop/1022677383


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References

  • [1] Aronson, D. G. (1968). Non-negative solutions of linear parabolic equations. Ann. Scuola Norm. Sup. Pisa 12 607-694.
  • [2] Berg, C., Christensen, J. P. R. and Ressel, P. (1984). Harmonic Analysis on Semigroups. Springer, New York.
  • [3] Dawson, D. A. (1993). Measure-valued Markov processes. Ecole d'Et´e de Probabilit´es de Saint-Flour XXI. Lecture Notes in Math. 1541 1-260. Springer, New York.
  • [4] Dynkin, E. (1993). Superprocesses and partial differential equations. Ann. Probab. 21 1185- 1262.
  • [5] Evans, S. N. and O'Connell, N. (1994). Weighted occupation time for branching particle systems and a representation for the supercritical superprocess. Canad. Math. Bull. 37 187-196.
  • [6] Fitzsimmons, P. J. (1988). Construction and regularity of measure-valued Markov branching processes. Israel J. Math. 64 337-361.
  • [7] Friedman, A. (1964). Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J.
  • [8] Iscoe, I. (1986). A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Related Fields 71 85-116.
  • [9] Marcus, M., Mizel, J. V. and Pinchover, Y. (1998). On the best constant for Hardy's inequality in Rd. Trans. Amer. Math. Soc. 350 3237-3255.
  • [10] Marcus, M. and Veron, L. (1997). Uniqueness and asymptotic behavior of solutions with boundary blow-up for a class of nonlinear elliptic equations. Ann. Inst. H. Poincar´e Anal. Nonlin´eaire 14 237-274.
  • [11] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.
  • [12] Pinchover, Y. (1996). On uniqueness and nonuniqueness of the positive Cauchy problem for parabolic equations with unbounded coefficients. Math.223 569-586.
  • [13] Pinsky, R. G. (1995). K-P-P-type asymptotics for nonlinear diffusion in a large ball with infinite boundary data and on Rd with infinite initial data outside a large ball. Comm. Partial Differential Equations 20 1369-1393.
  • [14] Pinsky, R. G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Univ. Press.
  • [15] Pinsky, R. G. (1996). Transience, recurrence and local extinction properties of the support for supercritical finite measure-valued diffusions. Ann. Probab. 24 237-267.
  • [16] Roelly-Coppoletta, S. (1995). A criterion of convergence of measure-valued processes: application to measure branching processes. Stochastics 17 43-65.