Derivative formula and applications for degenerate diffusion semigroups (Q387984)

The authors consider the SDE NEWLINE\[NEWLINE dX_t = Z(X_t) dt + (0,\sigma dB_t), \quad X_0=x\in \mathbb{R}^{m+d}, NEWLINE\]NEWLINE on \(\mathbb{R}^{m}\times\mathbb{R}^{d}\), where \(\sigma\) is an invertible \(d\times d\) matrix and \(B_t\) is a \(d\)-dimensional Brownian motion.NEWLINENEWLINE By using the Malliavin calculus and solving a control problem, the authors establish Bismut type derivative formulae for the associated Markov semigroup NEWLINE\[NEWLINE P_tf(x) = \mathbb{E}f(X_t(x)), \quad t>0,\; x\in \mathbb{R}^{m+d},\; f\in \mathcal{B}_b(\mathbb{R}^{m+d}), NEWLINE\]NEWLINE where \(\mathcal{B}_b(\mathbb{R}^{m+d})\) is the set of all bounded measurable functions on \(\mathbb{R}^{m+d}\).NEWLINENEWLINE As applications, they derive explicit gradient estimates and Harnack inequalities.