  The presence of geometric phases is known to affect the dynamics of the
systems involved. Here we consider a quantum degree of freedom, moving in a
dissipative environment, whose dynamics is described by a Langevin equation
with quantum noise. We show that geometric phases enter the stochastic noise
terms. Specifically, we consider small ferromagnetic particles (nano-magnets)
or quantum dots close to Stoner instability, and investigate the dynamics of
the total magnetization in the presence of tunneling coupling to the metallic
leads. We generalize the Ambegaokar-Eckern-Sch\"on (AES) effective action and
the corresponding semiclassical equations of motion from the U(1) case of the
charge degree of freedom to the SU(2) case of the magnetization. The Langevin
forces (torques) in these equations are strongly influenced by the geometric
phase. As a first but nontrivial application we predict low temperature quantum
diffusion of the magnetization on the Bloch sphere, which is governed by the
geometric phase. We propose a protocol for experimental observation of this
phenomenon.
