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 -(-△) 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.
Khoa LÊ
. LONG-TIME ASYMPTOTIC OF STABLE DAWSON-WATANABE PROCESSES IN SUPERCRITICAL REGIMES[J]. Acta mathematica scientia, Series B, 2019
, 39(1)
: 37
-45
.
DOI: 10.1007/s10473-019-0104-y
[1] Adamczak R, Miłoś P. CLT for Ornstein-Uhlenbeck branching particle system. Electron J Probab, 2015, 20:Art 42, 1-35
[2] 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
[3] Kouritzin M A, Le K. Long-time limits and occupation times for stable Fleming-Viot processes with decaying sampling rates. arXiv:1705.10685(2017)
[4] 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
[5] Kouritzin M A, Ren Y-X. A strong law of large numbers for super-stable processes. Stochastic Process Appl, 2014, 124(1):505-521
[6] Perkins E. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 2002:125-324
[7] Walsh J B. An Introduction to Stochastic Partial Differential Equations. New York:Springer-Verlag, 1986:265-439
