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Differentiability of semigroups is useful for many applications. Here wefocus on stochastic differential equations whose diffusion coefficient is thesquare root of a differentiable function but not differentiable itself. Forevery $m\in\{0,1,2\}$ we establish an upper bound for a $C^m$-norm of thesemigroup of such a diffusion in terms of the $C^m$-norms of the driftcoefficient and of the squared diffusion coefficient. The constants in ourupper bound are often dimension-independent. Our estimates are thus suitablefor analyzing certain high-dimensional and infinite-dimensional degeneratestochastic differential equations.

Differentiability of semigroups of stochastic differential equations with H\"older-continuous diffusion coefficients