Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form

By Marco Di Francesco, Sergio Polidoro. Advances in Differential Equations, 2006

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We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/ba985f368fe56460084557222c16c7da03271bc2

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In Applied Mathematics

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The Schauder estimate for kinetic integral equations (2018, methodology): “nce of the more general theory of ultraparabolic equations of Kolmogorov type. Results involving regularity estimates in Ho¨lder spaces for these equations appeared especially in the late 1990’s. See [22, 13, 12, 4, 2, 15], and the survey article [18]. More recently in [7, 9], Schauder estimates wer”
On conditional McKean Lagrangian stochastic models (2011, background): “We now recall the following estimate (see [6]): there exist η > 0 and c > 0, depending only on λ, T , and d such that (s, y, v; t,x, u) ≥ c η(s, y, v; t,x, u), ∀s < t < T, where η is defined in A.”