Reciprocal Diffusions and Symmetries of Parabolic PDE: The Nonflat Case |
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Authors: | Thieullen Michèle |
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Institution: | (1) Laboratoire de Probabilités et Modèles Aléatoires, UMR 7599, Boîte 188, Université Paris 6, 4, Place Jussieu, 75252 Paris Cedex 05, France |
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Abstract: | We introduce a new set of Reciprocal Characteristics for the class of reciprocal diffusions naturally associated to a general parabolic second-order linear differential operator. All the coefficients of this operator, including the diffusion matrix, depend on time. This set of reciprocal characteristics is provided by the study of the symmetries of the differential operator. The Riemannian metric defined by the diffusion matrix is of central importance. Our reciprocal characteristics are the natural extension of Ovsiannikov's differential invariants to the time dependent parabolic case. We also show that the symmetries of the PDE coincide with the one parameter families of transformations which leave the usual stochastic Lagrangian as well as a modified Onsager–Machlup Lagrangian invariant. |
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Keywords: | reciprocal diffusions symmetries second-order linear PDE one parameter group of transformations reciprocal characteristics invariance of a Lagrangian |
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