Jonas Tölle, Variability of paths and differential equations with BV-coefficients

Abstract :

In stochastic analysis, it is well-established to interpret stochastic differential equations (SDEs) in integrated form, a viewpoint conceptually strongly related to the distributional formulation of partial differential equations. However, there are many situations, where even the concept of the integral is subtle. Several powerful theories have emerged to treat these situations, such as rough path theory or the theory of regularity structures. On the other hand, these methods are usually applied to situations where the coefficient maps are smooth, and most of the existing methods break down completely if one allows discontinuities (provided that the forcing term is not too regular). In particular, this is the case if we admit general functions of bounded variation ($BV$-functions) as a possible choice of our nonlinear coefficients.

In this talk, we combine tools from fractional calculus and harmonic analysis, together with certain fine properties of $BV$-functions, allowing us to give a meaningful definition for (multidimensional) generalized Lebesgue-Stieltjes integrals for sufficiently regular Hölder functions. The key idea is that the unknown function should not spend too much time on the "bad" regions of the $BV$-coefficient maps. Our novel multiplicative composition estimate leads to a systematic way to quantify this in terms of potential theory of Riesz energies and the occupation measure of the unknown function. We discuss several consequences, and provide existence and uniqueness results for certain differential systems involving $BV$-coefficients, which can be applied to pathwise SDEs with focus on the fractional Brownian motion. Furthermore, we may relax our hypotheses by formulating them in terms of fractional Sobolev norms such that our results can be extended to certain discontinuous paths, such as typical realizations of certain Lévy processes.

The talk is based on joint works together with :

Michael Hinz (Bielefeld University) & Lauri Viitasaari (Uppsala University)