Derivative Formula and Gradient Estimates for Gruschin Type Semigroups

الأصل · EN

By solving a control problem and using Malliavin calculus, explicit derivative formula is derived for the semigroup Pₜ generated by the Gruschin type operator on ᵐ× ᵈ: L (x,y)= 1 2 {∑ᵢ₌₁ᵐ ₓᵢ² +∑ⱼ,ₖ₌₁ᵈ ((x)(x)*)jk ⱼₖ},(x,y)∈ ᵐ×ᵈ, where ∈ C¹(ᵐ; ᵈ⊗ᵈ) might be degenerate. In particular, if (x) is comparable with |x|ˡId× d for some l≥ 1 in the sense of (A4), then for any p>1 there exists a constant Cₚ>0 such that | Pₜ f(x,y)|≤ Cₚ (Pₜ |f|ᵖ)¹/ᵖßt ßt(|x|²+t)ˡ,t>0, f∈ (ᵐ⁺ᵈ), (x,y)∈ ᵐ⁺ᵈ, which implies a new Harnack type inequality for the semigroup. A more general model is also investigated.