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We study the long time behavior of a Brownian particle moving in ananomalously diffusing field, the evolution of which depends on the particleposition. We prove that the process describing the asymptotic behaviour of theBrownian particle has bounded (in time) variance when the particle interactswith a subdiffusive field; when the interaction is with a superdiffusive fieldthe variance of the limiting process grows in time as t^{2{\gamma}-1}, 1/2 <{\gamma} < 1. Two different kinds of superdiffusing (random) environments areconsidered: one is described through the use of the fractional Laplacian; theother via the Riemann-Liouville fractional integral. The subdiffusive field ismodeled through the Riemann-Liouville fractional derivative.

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