BEGIN:VCALENDAR
VERSION:2.0
PRODID:-//Talks.cam//talks.cam.ac.uk//
X-WR-CALNAME:Talks.cam
BEGIN:VEVENT
SUMMARY:Should I Stay Or Should I Go? Zero-Size Jumps In Random Walks For
  Lévy Flights - Gianni Pagnini (BCAM - Basque Center for Applied Mathemat
 ics)
DTSTART:20220224T113000Z
DTEND:20220224T120000Z
UID:TALK167792@talks.cam.ac.uk
DESCRIPTION:BCAM&nbsp\;&ndash\;&nbsp\;Basque Center for Applied Mathematic
 s\, Bilbao\, Spain &&nbsp\;Ikerbasque\, Bilbao\, Spain\nMotivated by the f
 act that\, in the literature dedicated to random walks for anomalous diffu
 sion\, it is disregarded if the walker does not move in the majority of th
 e iterations because the most frequent jump-size is zero (i.e.\, the jump-
 size distribution is unimodal with mode located in zero) or\, in oppositio
 n\, if the walker always moves because the jumps with zero-size never occu
 r (i.e.\, the jump-size distribution is bi-modal and equal to zero in zero
 )\, we provide an example in which indeed the shape of the jump-distributi
 on plays a role.\nIn particular\, we show that the convergence of Markovia
 n continuous-time random walk (CTRW) models for L&eacute\;vy flights to a 
 density function that solves the fractional diffusion equation is not guar
 anteed when the jumps follow a bi-modal power-law distribution equal to ze
 ro in zero\, but\, as a matter of fact\, the resulting diffusive process c
 onverges to a density function that solves a double-order fractional diffu
 sion equation [1].&nbsp\;Within this framework\, self-similarity is lost. 
 The consequence of this loss of self-similarity is the emergence of a time
 -scale for realising the large-time limit. Such time-scale results to span
  from zero to infinity accordingly to the power-law displayed by the tails
  of the walker&rsquo\;s density function. Hence\, the large-time limit cou
 ld not be reached in real systems.&nbsp\;\nThe significance of this result
  is two-fold: i) with regard to the probabilistic derivation of the fracti
 onal diffusion equation [2] and also ii) with regard to recurrence [3] and
  the related concept of site fidelity in the framework of L&eacute\;vy-lik
 e motion for wild animals [4].\nReferences\n[1] G. Pagnini and S. Vitali. 
 Should I stay or should I go? Zero-size jumps in random walks for L&eacute
 \;vy flights\,&nbsp\;Fract. Calc. Appl. Anal.\, 24(1)\, 137&ndash\;167\, 2
 021.&nbsp\;\n[2] E. Valdinoci. From the long jump random walk to the fract
 ional Laplacian.&nbsp\;Bol. Soc. Esp. Mat. Apl.\, 49\, 33&ndash\;44\, 2009
 .&nbsp\;\n[3] E. Affili\, S. Dipierro and E. Valdinoci\, Decay estimates i
 n time for classical and anomalous diffusion. In:&nbsp\;2018 MATRIX Annals
 &nbsp\;(D. Wood\, J. de Gier\, C. Praeger\, T. Tao\, Eds.)\, MATRIX Book S
 eries\, Vol. 3\, Springer\, Cham (2020)\, 167&ndash\;182.\n[4] R. Klages. 
 Search for food of birds\, fish and insects. In:&nbsp\;Diffusive Spreading
  in Nature\, Technology and Society&nbsp\;(A. Bunde\, J. Caro\, J. Kaerger
 \, G. Vogl\, Eds.)\, Springer\, Cham (2018)\, 49&ndash\;69.&nbsp\;
LOCATION:Seminar Room 1\, Newton Institute
END:VEVENT
END:VCALENDAR