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Abstract: We study the long time asymptotics of probability density functions pdfs ofL\-{e}vy flights in different confining potentials. For that we use two models:Langevin - driven and L\-{e}vy - Schr\-odinger semigroup - driven dynamics.It turns out that the semigroup modeling provides much stronger confiningproperties than the standard Langevin one. Since contractive semigroups set alink between L\-{e}vy flights and fractional pseudo-differential Hamiltoniansystems, we can use the latter to control the long - time asymptotics of thepertinent pdfs. To do so, we need to impose suitable restrictions upon theHamiltonian and its potential. That provides verifiable criteria for aninvariant pdf to be actually an asymptotic pdf of the semigroup-drivenjump-type process. For computational and visualization purposes ourobservations are exemplified for the Cauchy driver and its response to externalpolynomial potentials referring to L\-{e}vy oscillators, with respect to bothdynamical mechanisms.

Autor: Piotr Garbaczewski, Vladimir Stephanovich

Fuente: https://arxiv.org/

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