On the integration of Ito equations with a random or a W-symmetry
arXiv:2311.14998 · doi:10.1063/5.0141333
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.
arXiv comment50 pages; to appear in Journal of Mathematical Physics
References in corpus (16)
- Nonequilibrium Markov processes conditioned on large deviations
- Symmetry of stochastic non-variational differential equations
- Path integrals and stochastic calculus
- Symmetries of Stochastic Differential Equations: a geometric approach
- Nonlinear inhomogeneous Fokker-Planck equation within a generalized Stratonovich prescription
- Reduction and reconstruction of stochastic differential equations via symmetries
- Time evolution towards q-Gaussian stationary states through unified Ito-Stratonovich stochastic equation
- Random Lie-point symmetries of stochastic differential equations
- Symmetries of Stochastic Differential Equations using Girsanov transformations
- W-Symmetries of Ito stochastic differential equations
- A note on symmetries of diffusions within a martingale problem approach
- Reduction and reconstruction of SDEs via Girsanov and quasi Doob symmetries
- Integrable Ito equations and properties of the associated Fokker-Planck equations
- Symmetry classification of scalar autonomous Ito stochastic differential equations with simple noise
- Integrable Ito equations with multiple noises
- Integration by parts formulas and Lie's symmetries of SDEs