arXiv subjects

Hannes Kern

Publications and source records attributed to Hannes Kern.

5 recordsLinked to original sources

arXiv2024

Lipschitz estimates in the Besov settings for Young and rough differential equations

We develop a set of techniques that enable us to effectively recover Besov rough analysis from p-variation rough analysis. Central to our approach are new metric groups, in which some objects in rough path theory that have been previously viewed as two-parameter can be considered as path increments. Furthermore, we develop highly precise Lipschitz estimates for Young and rough differential equations, both in the variation and Besov scale.

math.PR↗

arXiv2024

A Stochastic Reconstruction Theorem on Rectangular Increments with an Application to a Mixed Hyperbolic SPDE

We extend the stochastic reconstruction theorem to a setting where the underlying family of distributions satisfies some natural conditions involving rectangular increments. This allows us to prove the well-posedness of a new class of mixed stochastic partial differential of hyperbolic type which combines standard Walsh stochastic integration and Young products.

math.PR↗

arXiv2023

Flow techniques for non-geometric RDEs on manifolds

In 2015, Bailleul presented a mechanism to solve rough differential equations by constructing flows, using the log-ODE method. We extend this notion in two ways: On the one hand, we localize Bailleul's notion of an almost-flow to solve RDEs on manifolds. On the other hand, we extend his results to non-geometric rough paths, living in any connected, cocommutative, graded Hopf algebra. This requires a new concept, which we call a pseudo bialgebra map. We further connect our results to Curry et al (2020), who solved planarly branched RDEs on homogeneous spaces.

math.PR↗

arXiv2023

A multiparameter Stochastic Sewing lemma and the regularity of local times associated to Gaussian sheets

We establish a multiparameter extension of the stochastic sewing lemma. This allows us to derive novel regularity estimates on the local time of locally non-deterministic Gaussian fields. These estimates are sufficiently strong to derive regularization by noise results for SDEs in the plain. In this context, we make the interesting and rather surprising observation that regularization effects profiting from each parameter of the underlying stochastic field in an additive fashion usually appear to be due to boundary terms of the driving stochastic field.

math.PR↗

arXiv2023

A stochastic reconstruction theorem

In a recent landmark paper, Khoa Lê (2020) established a stochastic sewing lemma which since has found many applications in stochastic analysis. He further conjectured that a similar result may hold in the context of the reconstruction theorem within Hairer's regularity structures. The purpose of this article is to provide such a stochastic reconstruction theorem. We also discuss two variations of this theorem, motivated by different constructions of stochastic integration against white noise. Our formulation makes use of the distributional viewpoint of Caravenna--Zambotti (2021).

math.PR↗