  Similarity solutions play an important role in many fields of science: we
consider here similarity in stochastic dynamics. Important issues are not only
the existence of stochastic similarity, but also whether a similarity solution
is dynamically attractive, and if it is, to what particular solution does the
system evolve. By recasting a class of stochastic PDEs in a form to which
stochastic centre manifold theory may be applied we resolve these issues in
this class. For definiteness, a first example of self-similarity of the
Burgers' equation driven by some stochastic forced is studied. Under suitable
assumptions, a stationary solution is constructed which yields the existence of
a stochastic self-similar solution for the stochastic Burgers' equation.
Furthermore, the asymptotic convergence to the self-similar solution is proved.
Second, in more general stochastic reaction-diffusion systems stochastic centre
manifold theory provides a framework to construct the similarity solution,
confirm its relevance, and determines the correct solution for any compact
initial condition. Third, we argue that dynamically moving the spatial origin
and dynamically stretching time improves the description of the stochastic
similarity. Lastly, an application to an extremely simple model of turbulent
mixing shows how anomalous fluctuations may arise in eddy diffusivities. The
techniques and results we discuss should be applicable to a wide range of
stochastic similarity problems.
