On the relation between transformations of SDEs and of associated Kolmogorov equations
Abstract
The relationship between transformations of one-dimensional stochastic differential equations and those of the associated Kolmogorov backward equations is investigated. Stochastic transformations combining deterministic time changes, space-time diffeomorphisms, and Girsanov-type changes of probability measure are shown to be in correspondence with admissible transformations of the associated linear parabolic equations, namely the transformations preserving the parabolic structure up to the possible introduction of a zero-order potential term. A distinguished subclass of transformations, characterized by the absence of any potential term in the transformed Kolmogorov equation, is shown to coincide exactly with Doob transformations of the underlying probability measure of the SDE. This characterization further establishes a correspondence between Doob symmetries of an SDE and Lie point symmetries of the associated Kolmogorov backward equation. This correspondence, in turn, yields a constructive (probabilistic) Lie symmetry classification of autonomous Kolmogorov backward equations.
Related
- Kinetic SDEs and kinetic Fokker-Planck equations with critical Morrey-type drifts, and local blow-ups
- On uniqueness of probability solutions to one-dimensional Fokker-Planck-Kolmogorov equations
- Fundamental solutions to Kolmogorov-Fokker-Planck equations with unbounded lower order terms
- Weighted derivative estimates for Kolmogorov equations
- Kolmogorov-Arnold Networks for Free-Boundary Partial Differential Equations
BibTeX
@misc{deho2026relation,
title = {{On the relation between transformations of SDEs and of associated Kolmogorov equations}},
author = {Susanna Dehò and Francesco Carlo De Vecchi and Paola Morando and Stefania Ugolini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.23482},
eprint = {2610.23482},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.23482},
}
RIS
TY - UNPB
TI - On the relation between transformations of SDEs and of associated Kolmogorov equations
AU - Susanna Dehò
AU - Francesco Carlo De Vecchi
AU - Paola Morando
AU - Stefania Ugolini
PY - 2026
DA - 2026/10/16
PB - arXiv
DO - 10.48550/arxiv.2610.23482
UR - https://arxiv.org/abs/2610.23482
AB - The relationship between transformations of one-dimensional stochastic differential equations and those of the associated Kolmogorov backward equations is investigated. Stochastic transformations combining deterministic time changes, space-time diffeomorphisms, and Girsanov-type changes of probability measure are shown to be in correspondence with admissible transformations of the associated linear parabolic equations, namely the transformations preserving the parabolic structure up to the possible introduction of a zero-order potential term. A distinguished subclass of transformations, characterized by the absence of any potential term in the transformed Kolmogorov equation, is shown to coincide exactly with Doob transformations of the underlying probability measure of the SDE. This characterization further establishes a correspondence between Doob symmetries of an SDE and Lie point symmetries of the associated Kolmogorov backward equation. This correspondence, in turn, yields a constructive (probabilistic) Lie symmetry classification of autonomous Kolmogorov backward equations.
ER -