Gaussian estimates for elliptic operators with unbounded drift

Abstract

We consider a strictly elliptic operator A u = ∑ i j D i ( a i j D j u ) − b ⋅ ∇ u + div ( c ⋅ u ) − V u , where 0 ⩽ V ∈ L loc ∞ , a i j ∈ C b 1 ( R N ) , b , c ∈ C 1 ( R N , R N ) . If div b ⩽ β V , div c ⩽ β V , 0 < β < 1 , then a natural realization of A generates a positive C 0 -semigroup T in L 2 ( R N ) . The semigroup satisfies pseudo-Gaussian estimates if | b | ⩽ k 1 V α + k 2 , | c | ⩽ k 1 V α + k 2 , where 1 2 ⩽ α < 1 . If α = 1 2 , then Gaussian estimates are valid. The constant α = 1 2 is optimal with respect to this property.