TY - JOUR

T1 - Mutually catalytic branching in the plane

T2 - Uniqueness

AU - Dawson, Donald A.

AU - Fleischmann, Klaus

AU - Mytnik, Leonid

AU - Perkins, Edwin A.

AU - Xiong, Jie

N1 - Funding Information:
∗Corresponding author. E-mail addresses: [email protected] (D.A. Dawson), [email protected] (K. Fleischmann), [email protected] (L. Mytnik), [email protected] (E.A. Perkins), [email protected] (J. Xiong). URLs: http://www.fields.utoronto.ca/dawson.html (D.A. Dawson), http://www.wias-berlin.de/~fleischm (K. Fleischmann), http://ie.technion.ac.il/leonid.phtml (L. Mytnik), http://www.math.ubc.ca/~perkins/perkins.html (E.A. Perkins), http://www.math.utk.edu/~jxiong/ (J. Xiong). 1Supported in part by an NSERC grant and a Max Planck Award. 2Supported in part by the DFG. 3Supported in part by the US–Israel Binational Science Foundation (grant No. 2000065) and the Israel Science Foundation (grant No. 116/01-10.0). 4Supported in part by an NSERC grant. 5Supported in part by UT-ORNL Science Alliance.

PY - 2003

Y1 - 2003

N2 - We study a pair of populations in ℝ 2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

AB - We study a pair of populations in ℝ 2 which undergo diffusion and branching. The system is interactive in that the branching rate of each type is proportional to the local density of the other type. Previous work had established the existence of such a process and derived some of its small scale and large scale properties. This paper is primarily focused on the proof of uniqueness of solutions to the martingale problem associated with the model. The self-duality property of solutions, which is crucial for proving uniqueness and was used in the previous work to derive many of the qualitative properties of the process, is also established.

KW - Catalytic super-Brownian motion

KW - Collision local time

KW - Duality

KW - Markov property

KW - Martingale problem

KW - Uniqueness

UR - http://www.scopus.com/inward/record.url?scp=0037267151&partnerID=8YFLogxK

U2 - 10.1016/S0246-0203(02)00006-7

DO - 10.1016/S0246-0203(02)00006-7

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AN - SCOPUS:0037267151

SN - 0246-0203

VL - 39

SP - 135

EP - 191

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 1

ER -