Long time asymptotics of a Brownian particle coupled with a random environment with non-diffusive feedback force

We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behavior of the Brownian particle has bounded (in time) variance when the particle interacts with a subdiffusive field; when the interaction is with a superdiffusive field the variance of the limiting process grows in time as t2γ−1, 1/2 < γ < 1. Two different kinds of superdiffusing (random) environments are considered: one is described through the use of the fractional Laplacian; the other via the Riemann–Liouville fractional integral. The subdiffusive field is modeled through the Riemann–Liouville fractional derivative. c ⃝ 2011 Elsevier B.V. All rights reserved.