Abstract
Let W = (Wt)t≥0 be a supercritical α-stable Dawson-Watanabe process (with α ∈ (0, 2]) and f be a test function in the domain of −(-Δ)α/2 satisfying some integrability condition. Assuming the initial measure W0 has a finite positive moment, we determine the long-time asymptotic of arbitrary order of Wt(f). In particular, it is shown that the local behavior of Wt in long-time is completely determined by the asymptotic of the total mass Wt(1), a global characteristic.
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References
Adamczak R, Mi loś P. CLT for Ornstein-Uhlenbeck branching particle system. Electron J Probab, 2015, 20: Art 42, 1–35
Asmussen S R, Hering H. Strong limit theorems for general supercritical branching processes with applications to branching diffusions. Z Wahrscheinlichkeitstheorie und Verw Gebiete, 1976, 36(3): 195–212
Kouritzin M A, Le K. Long-time limits and occupation times for stable Fleming-Viot processes with decaying sampling rates. arXiv:1705.10685 (2017)
Kouritzin M A, Le K, Sezer D. Laws of large numbers for supercritical branching Gaussian processes. Stochastic Process Appl, 2018. DOI: 10.1016/j.spa.1018.09.011
Kouritzin M A, Ren Y-X. A strong law of large numbers for super-stable processes. Stochastic Process Appl, 2014, 124(1): 505–521
Perkins E. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 2002: 125–324
Walsh J B. An Introduction to Stochastic Partial Differential Equations. New York: Springer-Verlag, 1986: 265–439
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The author thanks PIMS for its support through the Postdoctoral Training Centre in Stochastics during the completion of the paper.
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Lê, K. Long-Time Asymptotic of Stable Dawson-Watanabe Processes in Supercritical Regimes. Acta Math Sci 39, 37–45 (2019). https://doi.org/10.1007/s10473-019-0104-y
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DOI: https://doi.org/10.1007/s10473-019-0104-y