Fractional Langevin type equations for white noise distributions

Abstract

Abstract In this paper, by applying the intertwining properties, we introduce the fractional powers of the number operator perturbed by generalized Gross Laplacians (infinite dimensional Laplacians), which are special types of the infinitesimal generators of generalized Mehler semigroups. By applying the intertwining properties and semigroup approach, we study the Langevin type equations associated with the infinite dimensional Laplacians and with white noise distributions as forcing terms. Then we investigate the unique solution of the fractional Langevin type equations associated with the Riemann-Liouville and Caputo time fractional derivatives, and the fractional power of the infinite dimensional Laplacians, for which we apply the intertwining properties again. For our purpose, we discuss the fractional integrals and fractional derivatives of white noise distribution valued functions.