Gaussian estimates for elliptic operators with unbounded drift

We consider a strictly elliptic operatorAu=∑ijDi(aijDju)−b⋅∇u+div(c⋅u)−Vu, where 0⩽V∈Lloc∞, aij∈Cb1(RN), b,c∈C1(RN,RN)b,c∈C1(RN,RN). If divb⩽βVdivb⩽βV, divc⩽βVdivc⩽βV, 0<β<10<β<1, then a natural realization of AA generates a positive C0C0-semigroup T   in L2(RN)L2(RN). The semigroup satisfies pseudo-Gaussian estimates if|b|⩽k1Vα+k2,|c|⩽k1Vα+k2, where 12⩽α<1. If α=12, then Gaussian estimates are valid. The constant α=12 is optimal with respect to this property.