Symmetry properties of one- and two-dimensional Fokker-Planck equations

Shtelen, W. M.; Stogny, Valery

doi:10.1088/0305-4470/22/13/002

Cited by 50 publications

(31 citation statements)

References 3 publications

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“…It was additionally discussed in a number of papers. See, e.g., [9,69,[71][72][73]. Nevertheless, the symmetry criterion in its present forms is not as constructive as Cherkasov's.…”

Section: Discussionmentioning

confidence: 97%

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Conservation Laws and Potential Symmetries of Linear Parabolic Equations

2007

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We carry out an extensive investigation of conservation laws and potential symmetries for the class of linear (1+1)-dimensional second-order parabolic equations. The group classification of this class is revised by employing admissible transformations, the notion of normalized classes of differential equations and the adjoint variational principle. All possible potential conservation laws are described completely. They are in fact exhausted by local conservation laws. For any equation from the above class the characteristic space of local conservation laws is isomorphic to the solution set of the adjoint equation. Effective criteria for the existence of potential symmetries are proposed. Their proofs involve a rather intricate interplay between different representations of potential systems, the notion of a potential equation associated with a tuple of characteristics, prolongation of the equivalence group to the whole potential frame and application of multiple dual Darboux transformations. Based on the tools developed, a preliminary analysis of generalized potential symmetries is carried out and then applied to substantiate our construction of potential systems. The simplest potential symmetries of the linear heat equation, which are associated with single conservation laws, are classified with respect to its point symmetry group. Equations possessing infinite series of potential symmetry algebras are studied in detail.Comment: 67 pages, minor correction

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“…It was additionally discussed in a number of papers. See, e.g., [9,69,[71][72][73]. Nevertheless, the symmetry criterion in its present forms is not as constructive as Cherkasov's.…”

Section: Discussionmentioning

confidence: 97%

“…A modern treatment of the subject is given in [51]. There exist also a number of papers rediscovering results of Lie and Ovsiannikov [45,51] partially (see, e.g., [10,19,43,68,69,[71][72][73][74]). …”

Section: Group Classificationmentioning

confidence: 99%

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

2007

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“…A modern treatment can be found in [19] (see also [21]). A number of papers are devoted specifically to symmetries of FP equation in one spatial dimension [4,5,[22][23][24]. There are no general studies for higher dimensions.…”

Section: Introductionmentioning

confidence: 99%

“…There are no general studies for higher dimensions. The existing results are limited to a special case of Kramers' equation for the diffusion matrix which is constant and degenerate [23] and FP equation with a constant and positive definite diffusion matrix [7]. Both papers are restricted to FP equations in two spatial dimensions.…”

Section: Introductionmentioning

confidence: 99%

On Lie Group Classification of a Scalar Stochastic Differential Equation

Kozlov

2021

JNMP

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“…Basically this equation describes Gaussian stochastic systems with a white spectrum. The symmetry properties of point transformations of these types of equations have also been extensively investigated [5][6][7][8][9][10]. In a previous publication [11] we have shown that Fokker-Planck type equations belong to five basic groups of point transformations.…”

Section: Introductionmentioning

confidence: 99%

Group of contact transformations: Symmetry classification of Fokker-Planck type equations

Rudra

1999

Pramana - J Phys

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Fokker-Planck type equations have been classified according to the groups of contact transformations to which they belong. It has been found that there are only five classes as in the case of groups of point transformations. We have also obtained the algebraic structures of the corresponding Lie algebras. However, there are isomorphies in their group properties. The corresponding basis sets of functionally independent invariants formed by the generators of these groups have also been obtained.

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Symmetry properties of one- and two-dimensional Fokker-Planck equations

Cited by 50 publications

References 3 publications

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

Conservation Laws and Potential Symmetries of Linear Parabolic Equations

On Lie Group Classification of a Scalar Stochastic Differential Equation

Group of contact transformations: Symmetry classification of Fokker-Planck type equations

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