Weighted derivative estimates for Kolmogorov equations

For diffusions whose coefficients are smooth, with at most linear growth and bounded derivatives of higher order, we establish explicit pointwise bounds on the spatial derivatives of the classical solution to the Cauchy problem with polynomially growing initial data. We also obtain explicit growth estimates for the derivatives of the SDE with respect to the initial data. The proofs combine Lyapunov functions and Markov semigroups on weighted spaces with Kunita’s theory of the differentiable dependence of SDE solutions on the initial data.