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SUMMARY:Homogenization of Lévy operators with asymmetric densities - Mari
 ko Arisawa\, University of Vienna
DTSTART:20090508T130000Z
DTEND:20090508T140000Z
UID:TALK18330@talks.cam.ac.uk
CONTACT:Carola-Bibiane Schoenlieb
DESCRIPTION:The homogenization of the  Lévy operator appears in various m
 ulti-scale models in finance\, economy\, biology\, etc\, which use jump-di
 ffision process and pure jump process.  The Lévy operator is the infinite
 smal generator of such a process\, in the following non-local form: \n\n$$
 \nLu(x)= - int (R power N) [u(x+z) - u(x) - Du(x).z] d q(z)\,\n$$\n\nwhere
  $q(.)$  the  Lévy density represents the distribution of the length of j
 umps. When the density is symmetric\, i.e. $q(z)=1/(|z| power (N+alpha))$ 
 ($alpha in [0\,2)$)\, the operator is known as the fractional power of the
  Laplacian. In applications\, the Lévy densities are asymmetric in usual.
  We use the framework of the viscosity solution to treat such problems. In
  order to solve the homogenization of the integro-differential equation wi
 th the  Lévy operator\, we derive the ergodic cell peoblem. We prove the 
 ergodicity of the jump-diffusion process and pure jump process in torus\, 
 by using the PDE method. The homegenization result is then proved rigorous
 ly.
LOCATION:MR14\,  Centre for Mathematical Sciences\, Wilberforce Road\, Cam
 bridge
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