Asymptotic behaviour of densities of stable semigroups of measures
Tóm tắt
We prove that densities of the measures in a strictly stable semigroup (h
t
) of symmetric measures on a homogeneous group, if they exist, have the following asymptotic behaviour:
$$\mathop {\lim |}\limits_{|x| \to \infty } x|^{Q + \alpha } \cdot h_1 (x) = k(\bar x),$$
where α is the characteristic exponent,
$$\bar x = |x|^{ - 1} x$$
, andk is the density of the Lévy measure associated to the semigroup. Moreover, if
$$k(\bar x) = 0$$
a more precise description is given.
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