  The field theoretic renormalization group is applied to a simple model of
random walk on a rough fluctuating surface. We consider the Fokker--Planck
equation for a particle in a uniform gravitational field. The surface is
modelled by the generalized Edwards--Wilkinson linear stochastic equation for
the height field. The full stochastic model is reformulated as a
multiplicatively renormalizable field theory, which allows for application of
the standard renormalization theory. The renormalization group equations have
several fixed points that correspond to possible scaling regimes in the
infrared range (long times, large distances); all the critical dimensions are
found exactly. As an example, the spreading law for particle's cloud is
derived. It has the form $R^2(t)\simeq t^{2/\Delta_{\omega}}$ with the exactly
known critical dimension of frequency $\Delta_{\omega}$ and, in general,
differs from the standard expression $R^2(t)\simeq t$ for ordinary random walk.
