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.
On the integration of Ito equations with a random or a W-symmetry / G. Gaeta. - In: JOURNAL OF MATHEMATICAL PHYSICS. - ISSN 0022-2488. - 64:12(2023), pp. 123504.1-123504.30. [10.1063/5.0141333]
On the integration of Ito equations with a random or a W-symmetry
G. Gaeta
2023
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.File | Dimensione | Formato | |
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