Lindblad evolution with subelliptic diffusion

Abstract

We consider classical/quantum correspondence in Lindblad evolution with jump operators for which the corresponding Fokker--Planck equation is subelliptic. This allows us to consider the physical model proposed by Zurek and Paz, and to extend some of the recent mathematical results of Hernandez, Ranard and Riedel, Galkowski and Zworski, and Li, where the diffusion term in the Fokker-Planck equation was assumed elliptic. We consider the case where the jump operators $\ell_j$ in the Lindbladian are linear functions of $x$, and place an assumption which implies that the H\"ormander condition holds for the resulting Fokker-Planck equation. By constructing a suitable parametrix for this equation we show that the semiclassical derivative estimates established for elliptic diffusion also hold in the subelliptic case, with global bounds in $L^p$ for all $1\le p\le \infty$.