  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 behaviour 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 t^{2{\gamma}-1}, 1/2 <
{\gamma} < 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.
