Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

2011/09/30 by Feng‐Yu Wang, Wang, Feng-Yu
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.1109.6738

openalex publication_date 2011/09/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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× \R^d: L (x,y)=\ff 1 2 \∑_i=1^m \pp_xi² +∑_j,k=1^d (\si(x)\si(x)^*)_jk \pp_yj\pp_yk\, (x,y)∈ \R^m×\R^d, where \si∈ C¹(\R^m; \R^d⊗\R^d) might be degenerate. In particular, if \si(x) is comparable with |x|^lI_{d× d} for some l≥ 1 in the sense of (\refA4), then for any p>1 there exists a constant C_p>0 such that |\nn P_t f(x,y)|≤ \ffC_p (P_t |f|^p)^1/pßt∧ ßt(|x|²+t)^l, tgt;0, f∈ \B_b(\R^m+d), (x,y)∈ \R^m+d, which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.

Citations

• Quasi-invariance for heat kernel measures on sub-Riemannian infinite-dimensional Heisenberg groups
• Derivative Formula and Applications for Degenerate Diffusion Semigroups

Related

• Log-Harnack Inequality for Gruschin Type Semigroups
• Derivative Formula and Harnack Inequality for Degenerate Functional SDEs
• Derivative Formula and Applications for Degenerate Diffusion Semigroups
• Derivative Formulae and Poincaré Inequality for Kohn-Laplacian Type Semigroups
• Harnack Inequality and Gradient Estimate for G-SDEs with Degenerate Noise