We define fractional Pearson diffusions [5,7,8] by non-Markovian time change
in the corresponding Pearson diffusions [1,2,3,4]. They are governed by the
time-fractional diffusion equations with polynomial coefficients depending
on the parameters of the corresponding Pearson distribution. We present the
spectral representation of transition densities of fractional Pearson
diffusions, which depend heavily on the structure of the spectrum of the
infinitesimal generator of the corresponding non-fractional Pearson
diffusion. Also, we present the strong solutions of the Cauchy problems
associated with heavy-tailed fractional Pearson diffusions and the
correlation structure of these diffusions [6] .
Continuous time random walks have random waiting times between particle
jumps. We define the correlated continuous time random walks (CTRWs) that
converge to fractional Pearson diffusions (fPDs) [9,10,11]. The jumps in
these CTRWs are obtained from Markov chains through the Bernoulli urn-scheme
model, Wright-Fisher model and Ehrenfest-Brillouin-type models. The jumps
are correlated so that the limiting processes are not L\'{e}vy but diffusion
processes with non-independent increments.
\bigskip
This is a joint work with M. Meerschaert (Michigan State University, USA),
I. Papic (University of Osijek, Croatia), N.Suvak (University of Osijek,
Croatia) and A. Sikorskii (Michigan State University and Arizona University,
USA).
References:
[1] Avram, F., Leonenko, N.N and Suvak, N. (2013), On spectral analysis of
heavy-tailed Kolmogorov-Pearson diffusion, \textit{Markov Processes and
Related Fields}, Volume 19, N 2 , 249-298
[2] Avram, F., Leonenko, N.N and Suvak, N., (2013), Spectral representation
of transition density of Fisher-Snedecor diffusion, \textit{Stochastics}, 85
(2013), no. 2, 346--369
[3] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four
moments theorems on Markov chaos, \textit{Annals of Probablity}, in press
[4] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for
the Fisher-Snendecor diffusion, \textit{Bernoulli}, Vol. 19, No. 5B,
2294-2329
[5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional
Pearson diffusions, \textit{Journal of Mathematical Analysis and Applications%
}, vol. 403, 532-546
[6] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation
Structure of Fractional Pearson diffusion, \textit{Computers and Mathematics
with Applications}, 66, 737-745
[7] Leonenko,N.N., Meerschaert,M.M., Schilling,R.L. and Sikorskii, A. (2014)
Correlation Structure of Time-Changed L\'{e}vy Processes, \textit{%
Communications in Applied and Industrial Mathematics}, Vol. 6 , No. 1, p.
e-483 (22 pp.)
[8] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017)
Heavy-tailed fractional Pearson diffusions, \textit{Stochastic Processes and
their Applications}, 127, N11, 3512-3535
[9] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated
continuous time random walks and fractional Pearson diffusions, \textit{%
Bernoulli}, Vol. 24, No. 4B, 3603-3627
[10] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019)
Ehrenfest-Brillouin-type correlated continuous time random walks and
fractional Jacoby diffusion, \textit{Theory Probablity and Mathematical
Statistics}, Vol. 99,123-133.
[11] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019)
Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod
topology, submitted
