Local densities for a class of degenerate diffusions

Abstract

We study a class of R d -valued continuous strong Markov processes that are generated, only locally, by an ultra-parabolic operator with coefficients that are regular w.r.t. the intrinsic geometry induced by the operator itself and not w.r.t. the Euclidean one. The first main result is a local Itô formula for functions that are not twice-differentiable in the classical sense, but only intrinsically w.r.t. to a set of vector fields, related to the generator, satisfying the Hörmander condition. The second main contribution, which builds upon the first one, is an existence and regularity result for the local transition density.

Sharp Estimates for Geman–Yor Processes and applications to Arithmetic Average Asian options

“…On the other hand, we prove upper and lower bounds for Γ in terms of quantities only depending on the constants λ and Λ appearing in (1.17). In a future study we plan to combine the bounds (1.3) with, either pure PDEs methods, or with the local results established in [37], to prove the existence of a fundamental solution of L under weaker regularity assumptions on a and b.…”

“…Concerning the operator L in its general form, we recall that existence and regularity result for the local transition density were established in the recent article of Lanconelli, Pagliarani and Pascucci [37], under the assumption that the coefficients a and b belong to some space of Hölder continuous functions.…”

Optimal regularity for degenerate Kolmogorov equations with rough coefficients

“…The quasi-norm (1.7), hereafter referred to as to intrinsic norm for A + Y , can be directly related to the scaling properties of the underlying diffusion process (cf. [4], [13]). For example, the intrinsic norm for the Langevin operator (1.2) reads as…”

Strong regularization by noise for kinetic SDEs

“…We refer the reader to [13,Remark 1.15] for a comparison with other notions of intrinsic Hölder spaces proposed in the literature. Using intrinsic Taylor formulas it is possible to derive the following lemma (see also [11,Theorem 1.3]).…”