Let W =(W_t)_(t≥0) be a supercritical a-stable Dawson-Watanabe process(withα∈(0,2]) and f be a test function in the domain of-(-△)^(α/2) satisfying some integrability condition. Assuming the initial measure W_0 has a finite positive moment, we determine the long-time asymptotic of arbitrary order of W_t(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 W_t(1), a global characteristic.
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