Symmetry of stochastic equations

Symmetry methods are by now recognized as one of the main tools to attack deterministic differential equations (both ODEs and PDEs); see e.g. [3, 5, 9, 16, 21, 22, 25, 26]. The situation is quite different for what concerns stochastic differential equations [1, 13, 14, 17, 20]: here, symmetry considerations are of course quite widely used by theoretical physicists (see e.g. [4, 15, 18] in the context of KPZ theory), but a rigorous and general theory comparable to the one developed for deterministic equation is still lacking, and the attention of the community working on symmetry methods should maybe be called to this. We would like to quote here early work by Misawa [19], mainly concerned with conservation laws for stochastic systems, and on the work by Arnold and Imkeller [1, 2] for normal forms of stochatic equations. In the following I will report on some work I have done – to a large extent in collaboration with N. Rodr ́ iguez Quintero – on symmetries of stochastic (Ito) equations, and how these compare with the symmetries of the associated diffusion (Fokker-Planck) equations [6, 8, 23, 24]; further details can be found in [12, 10]. Part of this was concerned with providing a suitable definition for symmetries of a stochastic equation, so I will also show that the symmetries at the basis of the construction of the KPZ equation are actually recovered within the frame developed here.