Intrinsic Taylor formula for Kolmogorov-type homogeneous groups

Abstract

We consider a class of ultra-parabolic Kolmogorov-type operators satisfying the Hörmander's condition. We prove an intrinsic Taylor formula with global and local bounds for the remainder given in terms of the norm in the homogeneous Lie group naturally associated to the differential operator.

Local densities for a class of degenerate diffusions

“…We also highlight the fact that Assumption 1.2 is not required in Theorem 1.3. Just like the classical Itô formula is based on the standard Taylor expansion, the cornerstone of (1.8) is a non-Euclidean Taylor formula, proved in [12] and [13] for functions in C 2,α B , that roughly states that…”

“…We point out that slightly different versions of such spaces were previously adopted in several works (see [6] and [10] among others). Here we use the definition given [12], which is basically the one required in order to prove an intrinsic Taylor formula where the remainder is in terms of the intrinsic quasi-norm.…”

Intrinsic expansions for averaged diffusion processes

“…Intrinsic Hölder and Sobolev spaces for Kolmogorov operators were studied by several authors, among others [9], [2], [30], [29], [33] and [31]. In this paper we use the intrinsic Hölder spaces C n,α B in Definition 2.1 below, as defined in [36] where the authors also proved a Taylor formula with reminder expressed in terms of the homogeneous norm induced by the operator (see Theorem 2.3 below). Deferring precise definitions and statements until Section 2, the n-th order intrinsic Taylor polynomial, centered at ζ = (s, ξ)…”

“…In this section we collect some known facts about the intrinsic geometry of Kolmogorov operators. We also recall the definition of intrinsic Hölder spaces and the Taylor formula recently proved in [36]. We consider the prototype Kolmogorov operator obtained by (1.3)-(1.7) with A 0 equal to a scalar (p 0 × p 0 )-matrix and a i ≡ 0, i = 1, .…”

“…In the scalar case, this phenomenon was already observed in [19] using Malliavin calculus techniques. Here, we provide a general and detailed analysis by employing the recent study of intrinsic functional spaces related to hypoelliptic Kolmogorov operators in [36]. Applications to finance are discussed, in the study of path-dependent derivatives (e.g.…”