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Local time and Tanaka formula for a multitype Dawson-Watanabe superprocess. (English) Zbl 1115.60052

Superprocesses are continuous-time Markov processes whose values are locally finite measures on a locally compact space \(S\). This paper is devoted to a study of the local time of a continuous superprocess that arises as the diffusion limit of multitype particle systems in \({\mathbb R}^d\) with critical branching and symmetric stable motions. Its values are finite positive measures on the space \(S = \{ 1, 2, \dots, k \} \times {\mathbb R}^d\), where \(k \geq 1\) is the number of types. Historically, E. B. Dynkin [in: Les processus stochastiques Astérisque 157–158, 141–171 (1988; Zbl 0659.60105)] defined the local time of a continuous superprocess as a stochastic integral and gave a criterion for existence of local time.
The authors prove the existence of local time for dimensions \(d < 2 \alpha\), where the parameter \(\alpha \in ( 0, 2]\) denotes the smallest of the stable motion exponents, and also derive a representation of the local time which is a multitype analogue of Tanaka formula-like representations obtained by R. J. Adler and M. Lewin [Stochastic Processes Appl. 41, No. 1, 45–67 (1992; Zbl 0754.60086)] and J. A. López-Mimbela and J. Villa [J.Math.Sci., New York 121, No. 5, 2653–2663 (2004; Zbl 1070.60068)] for the super-Brownian local time. Actually this representation is used to show that Dynkin’s notion of superprocess local time coincides with the occupation density of the multitype superprocess when it makes sense.
Note that their result on existence of local time relies basically on Dynkin’s criterion of Theorem 1.4 (loc. cit.), and also that the representation of local time follows essentially from the martingale problem for the superprocess. However, technically speaking, an extension of the validity of the martingale problem to a wider class of cylindric functions is inevitable. This extension allows them to obtain the Tanaka formula-like representation by an \(L^2\)-approximation procedure.

MSC:

60G57 Random measures
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60J55 Local time and additive functionals
60G52 Stable stochastic processes
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References:

[1] Adler, Stochastic Process. Appl. 41 pp 45– (1992)
[2] Blumental, Trans. Amer. Math. Soc. 95 pp 263– (1960)
[3] Dynkin, Astérisque 157–158 pp 147– (1988)
[4] Gorostiza, Adv. in Appl. Probab. 22 pp 49– (1990)
[5] Spatial Branching Processes, Random Snakes and Partial Differential Equations, Lectures in Mathematics ETH Zürich (Birkhäuser, Basel, 1999).
[6] López-Mimbela, J. Math. Sci., New York 121 pp 2653– (2004)
[7] Representaciones Tipo Fórmula de Tanaka del Tiempo Local de Superprocesos, Ph. D. thesis, CIMAT, México (2002).
[8] Functional Analysis, sixth edition (Springer-Verlag, 1968).
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