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Non-local Pearson diffusions

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FD2W01 - Deterministic and stochastic fractional differential equations and jump processes

We define non-local  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 polynomial coefficients depending on the parameters of the corresponding Pearson distribution. We present the spectral representation of transition densities of non-local Pearson diffusions, which depend heavily on the structure of the spectrum of the infinitesimal generator of the corresponding Markovian Pearson diffusion. We focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker-Planck operator 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 diffusions by inverse subordinator. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence. In the case of inverse stable subordinator we define the correlated continuous time random walks (CTRWs) that converge to fractional Pearson diffusions (fPDs) [7,8,9]. The jumps in these CTR Ws are obtained from Markov chains through the Bernoulli urn-scheme model, Wright-Fisher model and Ehrenfest-Brillouin-type. This 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  (Michigan State University, USA ), and I. Papic and N.Suvak (University of Osijek, Croatia).     References:     [1] Ascione, G., Leonenko, N. and  Pirozzi, E. (2021) Time-Non-Local Pearson Diffusions, Journal of Statistical Physics, 183, N3, Paper No. 48     [2] Bourguin, S., Campese, S., Leonenko, N. and Taqqu,M.S. (2019) Four moments theorems on Markov chaos, Annals of Probability, 47, N3, 1417–1446     [3] Kulik, A.M. and Leonenko, N.N. (2013) Ergodicity and mixing bounds for the Fisher-Snedecor diffusion, Bernoulli, Vol. 19, No. 5B, 2294-2329     [4] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Fractional Pearson diffusions, Journal of Mathematical Analysis and Applications, vol. 403, 532-546     [5] Leonenko, N.N., Meerschaert, M.M and Sikorskii, A. (2013) Correlation Structure of Fractional Pearson diffusion, Computers and Mathematics with Applications, 66, 737-745     [6] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2017) Heavy-tailed fractional Pearson diffusions, Stochastic Processes and their Applications, 127, N11 , 3512-3535     [7] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2018) Correlated continuous time random walks and fractional Pearson diffusions, Bernoulli, Vol. 24, No. 4B, 3603-3627     [8] Leonenko, N.N., Papic, I., Sikorskii, A. and Suvak, N. (2019) Ehrenfest-Brillouin-type correlated continuous time random walks and fractional Jacoby diffusion, Theory Probability and Mathematical Statistics, Vol. 99,123-133.     [9] Leonenko, N. N.; Papić, I.; Sikorskii, A.; Šuvak, N. (2020) Approximation of heavy-tailed fractional Pearson diffusions in Skorokhod topology, Journal of Mathematical Analysis and Applications,  no. 2, 123934, 22 pp

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