Agnieszka Zięba
supervisor: Jacek Wesołowski
Quadratic harnesses are time-inhomogeneous Markov polynomial processes with linear conditional expectations and quadratic and linear conditional variances with respect to the past-future filtrations. Typically they are determined by five numerical constants hidden in the form of conditional variances. Well-known examples of quadratic harnesses are Wiener, Poisson or Gamma processes. This class includes also classical versions of the free Brownian motion, q-Gaussian process and q-Lévy-Meixner process.
In our work we derive infinitesimal generators of such processes, extending earlier known results, which are identified through a solution of a q-commutation equation in the algebra of infinite sequences of polynomials in one variable. The solution is a special element, whose coordinates satisfy a three terms recurrence and thus define a system of orthogonal polynomials. It turns out that the respective orthogonality measure μ uniquely determines the infinitesimal generator (acting on polynomials or bounded functions with bounded second derivative) as an integro-differential operator with explicit kernel, where the integration is with respect to the measure μ. Such formulas for infinitesimal generators of quadratic harnesses are of special
interest due to their relation to the ASEP (asymmetric simple exclusion process), one of the most extensively studied stochastic particle models nowadays.