Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

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 ${\mathbb{R}}^m\times {\mathbb{R}}^d:$ $$L(x,y)=\text{ff}12\{\sum _{i=1}^m{\text{pp}}_{x_i}^2+\sum _{j,k=1}^d(\text{si}(x)\text{si}(x{)}^{\ast }{)}_{jk}{\text{pp}}_{y_j}{\text{pp}}_{y_k}\},(x,y)\in {\mathbb{R}}^m\times {\mathbb{R}}^d,$$ where $\text{si}\in C^1({\mathbb{R}}^m;{\mathbb{R}}^d\otimes {\mathbb{R}}^d)$ might be degenerate. In particular, if $\text{si}(x)$ is comparable with $|x|^{l}I_{d\times d}$forsome$l\ge 1$inthesenseof(\text{ref}A4),thenforany$p>1$ there exists a constant $C_p>0$ such that $$\mathrm{\mid }\text{nn}{P}_{t}f(x,y)\mathrm{\mid }\le \text{ff}{C}_p(P_t\mathrm{\mid }f{\mathrm{\mid }}^p{)}^{1\mathrm{/}p}\text{ss}t\wedge \text{ss}t(\mathrm{\mid }x{\mathrm{\mid }}^2+t{)}^l,t>0,f\in {\text{B}}_b({\mathbb{R}}^{m+d}),(x,y)\in {\mathbb{R}}^{m+d},$$ which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.