  Fluctuations in a stellar system's gravitational field cause the orbits of
stars to evolve. The resulting evolution of the system can be computed with the
orbit-averaged Fokker-Planck equation once the diffusion tensor is known. We
present the formalism that enables one to compute the diffusion tensor from a
given source of noise in the gravitational field when the system's dynamical
response to that noise is included. In the case of a cool stellar disc we are
able to reduce the computation of the diffusion tensor to a one-dimensional
integral. We implement this formula for a tapered Mestel disc that is exposed
to shot noise and find that we are able to explain analytically the principal
features of a numerical simulation of such a disc. In particular the formation
of narrow ridges of enhanced density in action space is recovered. As the
disc's value of Toomre's $Q$ is reduced and the disc becomes more responsive,
there is a transition from a regime of heating in the inner regions of the disc
through the inner Lindblad resonance to one of radial migration of
near-circular orbits via the corotation resonance in the intermediate regions
of the disc. The formalism developed here provides the ideal framework in which
to study the long-term evolution of all kinds of stellar discs.
