Derivative formulas and Poincaré inequality for Kohn-Laplacian type semigroups

Abstract

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L :for σ an invertible m × m-matrix and {A l } 1≤l≤d some m × m-matrices such that the Hörmander condition holds. We first establish Bismut-type and Driver-type derivative formulae with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.

Bismut formula for Lions derivative of distribution dependent SDEs and applications

“…) has been well studied in the literature, where M x T is an integrable random variable on R d , which is measurable in x ∈ R d when it varies, see for instance [1,15,25,26,28] and references within. Since the coefficients are distribution independent, we have (1.12)…”

Estimates for invariant probability measures of degenerate SPDEs with singular and path-dependent drifts

“…Obviously, 1 ∈ H 1,2 σ (µ 0 ) with E 0 (1, 1) = 0. Moreover, according to [14,Corollary 1.2], there exists a universal constant c > 0 such that the log-Sobolev inequality (1.3) holds for κ = t 0 c and β = 0 (see [31] for more results on functional inequalities). So, assumption (H) holds.…”

Gradient bounds for Kolmogorov type diffusions

“…In [52][53][54] it has been shown that if we assume Ric (M ) K for some K ∈ R, then there exists a Markovian coupling of Brownian motions (B t ) t 0 and B t t 0 on M starting at p and p such that…”

“…This construction is known as a coupling by parallel transport. This coupling can be constructed using stochastic differential equations as in [24,54], or by a central limit theorem argument for the geodesic random walks as in [52]. It turns out that the existence of the coupling satisfying (4.14) is equivalent to…”