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Theory of Probability and Mathematical Statistics
(Teoriya Imovirnostei ta Matematychna Statystyka)



Ehrenfest‒Brillouin-type correlated continuous time random walk and fractional Jacobi diffusion

N. N. Leonenko, I. Papić, A. Sikorskii, N. Šuvak

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Abstract: Continuous time random walks (CTRWs) have random waiting times between particle jumps. Based on Ehrenfest–Brillouin-type model motivated by economics, we define the correlated CTRW that converge to the fractional Jacobi diffusion Y(E(t)), t≥0, defined as a time change of Jacobi diffusion process Y (t) to the inverse E(t) of the standard stable subordinator. In the CTRW considered in this paper, the jumps are correlated so that in the limit the outer process Y(t) is not a Lévy process but a diffusion process with non-independent increments. The waiting times between jumps are selected from the domain of attraction of a stable law, so that the correlated CTRWs with these waiting times converge to Y(E(t)).

Keywords: Correlated continuous time random walk, Ehrenfest–Brillouin Markov chain, fractional diffusion, Jacobi diffusion, Pearson diffusion.

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