  An appealing feature of inflationary cosmology is the presence of a
phase-space attractor, "slow roll", which washes out the dependence on initial
field velocities. We investigate the robustness of this property under
backreaction from quantum fluctuations using the stochastic inflation formalism
in the phase-space approach. A Hamiltonian formulation of stochastic inflation
is presented, where it is shown that the coarse-graining procedure - where
wavelengths smaller than the Hubble radius are integrated out - preserves the
canonical structure of free fields. This means that different sets of canonical
variables give rise to the same probability distribution which clarifies the
literature with respect to this issue. The role played by the
quantum-to-classical transition is also analysed and is shown to constrain the
coarse-graining scale. In the case of free fields, we find that quantum
diffusion is aligned in phase space with the slow-roll direction. This implies
that the classical slow-roll attractor is immune to stochastic effects and thus
generalises to a stochastic attractor regardless of initial conditions, with a
relaxation time at least as short as in the classical system. For non-test
fields or for test fields with non-linear self interactions however, quantum
diffusion and the classical slow-roll flow are misaligned. We derive a
condition on the coarse-graining scale so that observational corrections from
this misalignment are negligible at leading order in slow roll.
