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Let P t be the semigroup of the diffusion process generated by L := Σ i,j a ij ∂ i ∂ j + Σ i b i ∂ i on R d . It is proved that there exists c ∈ R and an R d -valued function b = (b i ) such that |⊇|P t f| ≤ e ct P t |⊇f| holds for all t > 0 and all f ∈ C 1 b (R d ) if and only if a = (a ij ) satisfies the formula ∂ k a ij + ∂ j a ki + ∂ i a kj = 0 for all i, j, k.

A character of the gradient estimate for diffusion semigroups