Optimal regularity for degenerate Kolmogorov equations with rough coefficients

Abstract

We consider a class of degenerate equations satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin-type diffusions.

Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation

“…In particular, the authors construct it and establish that is has the appropriate regularity and Gaussian bounds. We also mention connections to the other very recent preprint by Lucertini, Pagliarani, and Pascucci [46] in which the authors deduce optimal bounds on the higher regularity of the fundamental solution. The estimates in [46] are strong enough to replace Lemma 2.2 in our proof of Theorem 1.1.…”

“…We also mention connections to the other very recent preprint by Lucertini, Pagliarani, and Pascucci [46] in which the authors deduce optimal bounds on the higher regularity of the fundamental solution. The estimates in [46] are strong enough to replace Lemma 2.2 in our proof of Theorem 1.1. It seems likely that one could establish Theorem 1.1 via an alternative approach to the Schauder estimates using the results of [46] directly.…”

“…The estimates in [46] are strong enough to replace Lemma 2.2 in our proof of Theorem 1.1. It seems likely that one could establish Theorem 1.1 via an alternative approach to the Schauder estimates using the results of [46] directly.…”

“…The authors would like to sincerely thank Marco Bramanti and Andrea Pascucci for helpful discussions regarding their preprints [7,46]. CH was partially supported by NSF grants DMS-2003110 and DMS-2204615.…”

Approximation and Interpolation in Kolmogorov-type groups

“…A regularity theory for equations with variable coefficients, built on this functional setting, is useful when one needs to consider coefficients or data that are irregular in time (see also [5], [22], [6], [16] for recent developments). This approach leads to mild or weak/distributional solutions also because it cannot benefit from any regularizing effect of the semigroup in time.…”