Speaker: Nenad Šuvak, Josip Juraj Strossmayer University of Osijek, Croatia
Title: Pearson diffusions beyond the SDE: time changes, boundaries, and spectra
Abstract: Pearson diffusions form a small but rich class of one-dimensional diffusions, characterised by a linear drift and an at most quadratic squared diffusion coefficient, whose invariant laws are the classical Pearson distributions. Three members of this class (inverse-gamma, Fisher–Snedecor and Student diffusions) have heavy-tailed invariant laws, only finitely many moments, and generators whose spectrum splits into a finite discrete part and an absolutely continuous part. This split is invisible in the coefficients of the defining stochastic differential equation, yet it governs transition densities, correlation decay and statistical tractability.
The talk follows the inverse-gamma diffusion through three modifications.
First, replacing deterministic time by the inverse of a stable subordinator destroys the Markov property, makes the governing equation fractional in time, and slows the decay of correlations from exponential to power-law; the resulting process arises as scaling limits of continuous-time random walk with correlated jumps.
Second, a specific functional transformation of a light-tailed Pearson diffusion (the Cox–Ingersoll–Ross process in the inverse-gamma case), observed on an absolutely continuous stochastic clock adapted to the base process, yields a heavy-tailed diffusion from the Pearson class. Unlike the first modification, it preserves the Markov property.
Third, truncating the state space to a bounded interval, with or without a specific state-dependent killing mechanism in the interior and with either absorption or reflection at the finite endpoint, removes the continuous spectrum altogether: the spectrum becomes purely discrete and simple regardless of the boundary condition imposed, and the transition density is recovered as a convergent eigenfunction expansion with explicit asymptotics of the eigenvalues.
The presentation and research visit are supported by the COST Action STOCHASTICA
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