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SUMMARY:Domains of generators of Levy-type processes - Rene Schilling (Tec
 hnische Universität Dresden)
DTSTART:20220223T110000Z
DTEND:20220223T113000Z
UID:TALK167768@talks.cam.ac.uk
DESCRIPTION:We study the domain of the generator of stable processes\, sta
 ble-like processes and more general pseudo- and integro-differential opera
 tors which naturally arise both in analysis and as infinitesimal generator
 s of L&eacute\;vy- and L&eacute\;vy-type (Feller) processes. In particular
  we obtain conditions on the symbol of the operator ensuring that certain 
 (variable order) H&ouml\;lder and H&ouml\;lder&ndash\;Zygmund spaces are i
 n the domain. We use toolsfrom probability theory to investigate the small
 -time asymptotics of the generalized moments of a L&eacute\;vy or L&eacute
 \;vy-type process\,\n$$\n&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;
 &nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\;&nbsp\; \\lim_{t\\to 0} (
 E^xf(X_t)-f(x))/t\n$$\nfor functions $f$ which are not necessarily bounded
  or differentiable. The pointwise limit exists for fixed if $f$ satisfies 
 a H&ouml\;lder condition at x. Moreover\, we give sufficient conditions wh
 ich ensure that the limit exists uniformly in the space of continuous func
 tions vanishing at infinity. Our results apply\, in particular\, to stable
 -like processes\, relativistic stable-like processes\, solutions of L&eacu
 te\;vy-driven SDEs and L&eacute\;vy processes.
LOCATION:Seminar Room 1\, Newton Institute
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