Fundamental solutions of nonlocal Hörmander's operators. II.
The author considers nonlocal integro-differential Hörmander operator with smooth coefficients and introduces uniform Hörmander-type conditions on the coefficients of this operator. Under these conditions, using \textit{J.-M. Bismut}'s [Z. Wahrscheinlichkeitstheor. Verw. Geb. 63, 147--235 (1983; Zbl 0494.60082)] approach to the Malliavin calculus with jumps, the existence of fundamental solutions to the Hörmander operator is proved. From the point of view of stochastic differential equations (SDEs), the problem is equivalent to the study of distributional density of the solution of SDE with \(d\)-dimensional Brownian motion and independent stable subordinator. The smoothness of the distributional density is established for nonconstant diffusion coefficient. For Part I, see [\textit{X. Zhang}, Commun. Math. Stat. 4, No. 3, 359--402 (2016; Zbl 1365.60064)].
nonlocal integro-differential Hörmander operator
scientific article; zbMATH DE number 6754787