64 research outputs found
Kato classes for L\'evy processes
We prove that the definitions of the Kato class by the semigroup and by the
resolvent of the L\'{e}vy process on coincide if and only if 0
is not regular for {0}. If 0 is regular for {0} then we describe both classes
in detail. We also give an analytic reformulation of these results by means of
the characteristic (L\'{e}vy-Khintchine) exponent of the process. The result
applies to the time-dependent (non-autonomous) Kato class. As one of the
consequences we obtain a simultaneous time-space smallness condition equivalent
to the Kato class condition given by the semigroup.Comment: 30 pages. We have shortened some argument
Hitting times of points and intervals for symmetric L\'{e}vy processes
For one-dimensional symmetric L\'{e}vy processes, which hit every point with
positive probability, we give sharp bounds for the tail function of the first
hitting time of B which is either a single point or an interval. The estimates
are obtained under some weak type scaling assumptions on the characteristic
exponent of the process. We apply these results to prove optimal estimates of
the transition density of the process killed after hitting B.Comment: 39 page
Dirichlet heat kernel for unimodal L\'evy processes
We estimate the heat kernel of the smooth open set for the isotropic unimodal
pure-jump L\'evy process with infinite L\'evy measure and weakly scaling
L\'evy-Kchintchine exponent.Comment: 38 page
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