20 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
Estimates of gradient perturbation series
We give upper and lower bounds of perturbation series for transition
densities, corresponding to additive gradient perturbations satisfying certain
space-time integrability conditions
Regularity of fundamental solutions for L\'evy-type operators
For a class of non-symmetric non-local L\'evy-type operators
, which include those of the form
\mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1}
\left)\kappa(x,z)J(z)\, dz\,, we prove regularity of
the fundamental solution to the equation