Speaker: Susanna Dehò (University of Milano "La Statale")
Title: Symmetries of SDEs: from invariance properties to integration by parts formulas
Abstract: The study of symmetries in differential equations, pioneered by Sophus Lie, constitutes a fundamental geometric approach to understand the invariance properties governing a dynamical system. By identifying the groups of transformations that preserve the equation’s structure, this framework provides systematic tools for order reduction, the construction of exact solutions, and the simplification of complex problems. Although symmetry analysis stands as a classical pillar for deterministic differential equations (ODEs and PDEs), supported by extensive literature, its extension to Stochastic Differential Equations (SDEs) represents a relatively recent field of research.
Recent literature has also highlighted significant connections with the symmetries of the associated Fokker-Planck or Kolmogorov equations. Furthermore, the application of Lie symmetry theory to SDEs enables the derivation of integration by parts formulas inspired by Bismut’s variational approach to Malliavin calculus, with notable applications to the analysis of the law and regularity of the processes, as well as to the development of a stochastic calculus of variations.
In this talk, we will discuss various notions of symmetry for SDEs, highlighting their connections to established invariance properties of well-known stochastic models and the symmetries of Kolmogorov PDEs. We will analyze how a geometric approach, inspired by Lie’s deterministic framework, enables the development of powerful computational tools for symmetry calculation. Subsequently, we will demonstrate how applying this geometric theory allows for the constructive derivation of an integration by parts formula for SDEs, rooted in Bismut’s approach. Finally, we will show how this integration by parts formula acts as a generating identity for well-known probability formulas, and discuss its connections to Stein’s identities.
This contribution is based on a joint work with F.C. De Vecchi, P. Morando and S. Ugolini.
Susanna Dehò, Francesco C. De Vecchi, Paola Morando, and Stefania Ugolini. Random rotational invariance of integration by parts formulas within a Bismut-type approach. Journal of Mathematical Physics: in print, 2025.
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