Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

TL;DR

This paper derives explicit derivative formulas and gradient estimates for Gruschin type semigroups using control and Malliavin calculus, leading to new Harnack inequalities even with degenerate diffusion coefficients.

Contribution

It provides explicit derivative formulas and gradient bounds for Gruschin semigroups with degenerate coefficients, extending previous results and establishing new Harnack inequalities.

Findings

Explicit derivative formula for Gruschin semigroups derived.

Gradient estimates established under degenerate conditions.

New Harnack inequalities proved for the semigroup.

Abstract

By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup $P_t$ generated by the Gruschin type operator on $\R^{m}\times \R^{d}:$ $$L (x,y)=\ff 1 2 \bigg\{\sum_{i=1}^m \pp_{x_i}^2 +\sum_{j,k=1}^d (\si(x)\si(x)^*)_{jk} \pp_{y_j}\pp_{y_k}\bigg\},\ \ (x,y)\in \R^m\times\R^d,$$ where $\si\in C^1(\R^m; \R^d\otimes\R^d)$ might be degenerate. In particular, if $\si(x)$ is comparable with $|x|^{l}I_{d\times d}$ for some $l\ge 1$ in the sense of (\ref{A4}), then for any $p>1$ there exists a constant $C_p>0$ such that $$|\nn P_t f(x,y)|\le \ff{C_p (P_t |f|^p)^{1/p}}{\ss{t}\land \ss{t(|x|^2+t)^l}},\ \ t>0, f\in \B_b(\R^{m+d}), (x,y)\in \R^{m+d},$$ which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.