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SUMMARY:Non-local Pearson diffusions - Nikolai  Leonenko (Cardiff Universi
 ty)
DTSTART:20220223T093000Z
DTEND:20220223T100000Z
UID:TALK167762@talks.cam.ac.uk
DESCRIPTION:We define non-local &nbsp\;Pearson diffusions [1\,4\,6\,7] by 
 non-Markovian time change in the corresponding Pearson diffusions [2\,3\,4
 ]. They are governed by the time-non-local diffusion equations with polyno
 mial coefficients depending on the parameters of the corresponding Pearson
  distribution. We present the spectral representation of transition densit
 ies of non-local Pearson diffusions\, which depend heavily on the structur
 e of the spectrum of the infinitesimal generator of the corresponding Mark
 ovian Pearson diffusion. We focus on strong solutions of some heat-like pr
 oblems with a non-local derivative in time induced by a Bernstein function
  and an elliptic operator given by the generator or the Fokker-Planck oper
 ator of a Pearson diffusion. In particular\, we use spectral decomposition
  results for the usual Pearson diffusion. Moreover\, we provide stochastic
  representation of such solutions in terms of time-changed Pearson diffusi
 ons by inverse subordinator. Finally\, we exploit some further properties 
 of these processes\, such as limit distributions and long/short-range depe
 ndence.\nIn the case of inverse stable subordinator we define the correlat
 ed continuous time random walks (CTRWs) that converge to fractional Pearso
 n diffusions (fPDs) [7\,8\,9]. The jumps in these CTRWs are obtained from 
 Markov chains through the Bernoulli urn-scheme model\, Wright-Fisher model
  and Ehrenfest-Brillouin-type.\nThis is joint results with G.Ascione and E
 .Pirozzi (University of Naples\, Italy). Some results used in the talk are
  obtained jointly with M. Meerschaert and A. Sikorskii &nbsp\;(Michigan St
 ate University\, USA)\, and I. Papic and N.Suvak (University of Osijek\, C
 roatia).\n&nbsp\;&nbsp\;&nbsp\; References:\n&nbsp\;&nbsp\;&nbsp\; [1] Asc
 ione\, G.\, Leonenko\, N. and&nbsp\; Pirozzi\, E. (2021) Time-Non-Local Pe
 arson Diffusions\, Journal of Statistical Physics\, 183\, N3\, Paper No. 4
 8\n&nbsp\;&nbsp\;&nbsp\; [2] Bourguin\, S.\, Campese\, S.\, Leonenko\, N. 
 and Taqqu\,M.S. (2019) Four moments theorems on Markov chaos\, Annals of P
 robability\, 47\, N3\, 1417&ndash\;1446\n&nbsp\;&nbsp\;&nbsp\; [3] Kulik\,
  A.M. and Leonenko\, N.N. (2013) Ergodicity and mixing bounds for the Fish
 er-Snedecor diffusion\, Bernoulli\, Vol. 19\, No. 5B\, 2294-2329\n&nbsp\;&
 nbsp\;&nbsp\; [4] Leonenko\, N.N.\, Meerschaert\, M.M and Sikorskii\, A. (
 2013) Fractional Pearson diffusions\, Journal of Mathematical Analysis and
  Applications\, vol. 403\, 532-546\n&nbsp\;&nbsp\;&nbsp\; [5] Leonenko\, N
 .N.\, Meerschaert\, M.M and Sikorskii\, A. (2013) Correlation Structure of
  Fractional Pearson diffusion\, Computers and Mathematics with Application
 s\, 66\, 737-745\n&nbsp\;&nbsp\;&nbsp\; [6] Leonenko\, N.N.\, Papic\, I.\,
  Sikorskii\, A. and Suvak\, N. (2017) Heavy-tailed fractional Pearson diff
 usions\, Stochastic Processes and their Applications\, 127\, N11\, 3512-35
 35\n&nbsp\;&nbsp\;&nbsp\; [7] Leonenko\, N.N.\, Papic\, I.\, Sikorskii\, A
 . and Suvak\, N. (2018) Correlated continuous time random walks and fracti
 onal Pearson diffusions\, Bernoulli\, Vol. 24\, No. 4B\, 3603-3627\n&nbsp\
 ;&nbsp\;&nbsp\; [8] Leonenko\, N.N.\, Papic\, I.\, Sikorskii\, A. and Suva
 k\, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random w
 alks and fractional Jacoby diffusion\, Theory Probability and Mathematical
  Statistics\, Vol. 99\,123-133.\n&nbsp\;&nbsp\;&nbsp\; [9] Leonenko\, N. N
 .\;&nbsp\;Papić\, I.\;&nbsp\;Sikorskii\, A.\;&nbsp\;&Scaron\;uvak\, N.&nb
 sp\;(2020) Approximation of heavy-tailed fractional Pearson diffusions in 
 Skorokhod topology\, Journal of Mathematical Analysis and Applications\, &
 nbsp\;no. 2\,&nbsp\;123934\, 22 pp
LOCATION:Seminar Room 1\, Newton Institute
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