On the integration of Ito equations with a random or a W-symmetry

Abstract

Symmetries can be used to integrate scalar Ito equation -- or reduce systems of such equations -- by the Kozlov substitution, i.e. passing to symmetry adapted coordinates. While the theory is well established for so called deterministic standard symmetries (the class originally studied by Kozlov), some points need clarification for so called random standard symmetries and W-symmetries. This paper is devoted to such clarification; in particular we note that the theory naturally calls, for these classes of symmetries, to also consider generalized Ito equations; and that while Kozlov theory is extended substantially unharmed for random standard symmetries, W-symmetries should be handled with great care, and cannot be used towards integration of stochastic equations, albeit they have different uses.

50 pages; to appear in Journal of Mathematical Physics

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• Symmetry of stochastic non-variational differential equations
• Path integrals and stochastic calculus
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Cited by in corpus (1)

• Integrable Ito equations and properties of the associated Fokker-Planck equations