Symmetry of stochastic non-variational differential equations

Abstract

I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more and less recent attempts to extend this theory to the study of stochastic differential equations, and briefly mention some perspective in this direction.Comment: Review paper (119 pages) here in bookstyle. The published version may differ from the present (preprint) one. Version 2 (November 2017): due to a mistake in a basic formula (in a previous paper of mine) the content of Section 5.5 gives wrong statement; an ERRATUM has been added at the end of the file; see also arXiv:1711.01999 for correct results on that topi

W-symmetries of Ito stochastic differential equations

“…standard deterministic symmetries, standard random ones, and W-symmetries. The first two types have been studied, by ourselves and different authors, in the literature [9,10,11,13,16,17,18,19,20,21,23,27], while W-symmetries had so far lacked attention.…”

“…Note that here we will raise and lower indices -also inside spatial derivatives -making use of the assumption we are working in an Euclidean space; it would be interesting to study if one obtains different results in a general Riemannian manifold 9. We recall that we are considering infinitesimal (near-identity) maps, see (58); hence we are dealing with the connected component of the identity in the group, and with generators.…”

Weak symmetries of stochastic differential equations driven by semimartingales with jumps

“…Since K is a flat connection, and since without lost of generality we can suppose that H is simply connected, there exists a unique function (B(x ′ 0 , y), η(x ′ 0 , y)) defined on H x ′ 0 and solving the equations (38) and (39) on H x ′ 0 . Indeed (B(x ′ 0 , y), η(x ′ 0 , y)) can be obtained as the parallel transport of the identity from the point S(x ′ 0 ) into the point y through the flat connection K. The flatness of the connection K ensures that B(x ′ 0 , y) and η(x ′ 0 , y) solve equations (38) and (39). By repeating this construction for any x ′ ∈ M ′ , since the section S is smooth, we obtain two smooth functions B(x ′ , y) and η(x ′ , y) on U = M ′ × H solving equations (38) and (39).…”

Erratum: “Random Lie-point symmetries of stochastic differential equations” [J. Math. Phys. 58, 053503 (2017)]

“…2 ) This error has no consequence on our general discussion-conducted in terms of the ∆ operatorexcept for Sec. VIII (see below); but it does affect the specific computations occurring in concrete examples and some side remarks.…”