Develops existence, uniqueness, and regularity theory for Itô equations and parabolic PDEs with singular drifts using Morrey-space conditions that are new even when the drift vanishes.
On the heat equation with singular drift
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abstract
We prove the maximum modulus estimates in terms of the $L_{q,p}$-norm of the free term for solutions of the heat equation with Morrey drift for any $q,p$ satisfying $d/p+2/q<2$ and any order of integration in the definition of the $L_{q,p}$-norm. An application to the case of $b$ satisfying the Ladyzhenskaya-Prodi-Serrin condition is given. The technique is easily adaptable to equations with Laplacians of order $\geq 1$.
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Stochastic It\^o Equations and Parabolic Second-Order Equations with singular Drift
Develops existence, uniqueness, and regularity theory for Itô equations and parabolic PDEs with singular drifts using Morrey-space conditions that are new even when the drift vanishes.