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On the parabolic Adams theorem and its applications to diffusion processes

N. V. Krylov

TL;DR

The paper develops a unified PDE/Stochastic-Analysis framework for parabolic Itô equations with divergence-form operators and low-regularity, possibly singular drift terms. By leveraging the parabolic Adams theorem and parabolic Riesz potentials, it obtains robust weighted $L_{p}$ estimates for the parabolic evolution family and moment bounds for derivatives with respect to initial data, under Morrey-class lower-order terms. It establishes Hölder regularity and strong-solution-type results for stochastic flows when $D\sigma$ and $b$ are small in parabolic Morrey spaces, and it presents a conjecture extending these results to drifts with stronger time-singularities, with a plan to prove via the developed gradient-estimate machinery. The work provides a principled path to weak- and strong-solution theory in settings with minimal coefficient regularity and to quantitative stability of stochastic flows under Morrey-type control.

Abstract

We show how the parabolic version of the Adams theorem and its corollary can be used to estimate in $L_{p}$ the evolution family associated to a divergence form second-order parabolic operator with parabolic Morrey lower-order terms and also how to estimate the moments of the derivatives of solutions of Itô equations with respect to the initial data when the drift term has singularities.

On the parabolic Adams theorem and its applications to diffusion processes

TL;DR

The paper develops a unified PDE/Stochastic-Analysis framework for parabolic Itô equations with divergence-form operators and low-regularity, possibly singular drift terms. By leveraging the parabolic Adams theorem and parabolic Riesz potentials, it obtains robust weighted estimates for the parabolic evolution family and moment bounds for derivatives with respect to initial data, under Morrey-class lower-order terms. It establishes Hölder regularity and strong-solution-type results for stochastic flows when and are small in parabolic Morrey spaces, and it presents a conjecture extending these results to drifts with stronger time-singularities, with a plan to prove via the developed gradient-estimate machinery. The work provides a principled path to weak- and strong-solution theory in settings with minimal coefficient regularity and to quantitative stability of stochastic flows under Morrey-type control.

Abstract

We show how the parabolic version of the Adams theorem and its corollary can be used to estimate in the evolution family associated to a divergence form second-order parabolic operator with parabolic Morrey lower-order terms and also how to estimate the moments of the derivatives of solutions of Itô equations with respect to the initial data when the drift term has singularities.
Paper Structure (7 sections, 13 theorems, 219 equations)

This paper contains 7 sections, 13 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Main results and discussion
  3. Parabolic Riesz potentials
  4. Proof of Theorem \ref{['theorem 11.26,3']}
  5. Regularity of solutions of stochastic equations as functions of initial data
  6. Proof of Theorem \ref{['theorem 6.21.1']}
  7. Proof of Theorem \ref{['theorem 3.7.1']}

Key Result

Theorem 2.1

Let $n\in\{iI_{i=1}+2iI_{i\geq2},i=1,2,...\}, \lambda\geq 0$. Then there are constants $\hat{{\sf a}},\hat{b}\in(0,1)$, depending only on $d,\delta,p_{0}$, $n$, such that, if $\hat{ {\sf a} }_{p_{0},\rho_{0}}\leq e^{- \lambda\rho_{0}}\hat{{\sf a}}$, $\hat{b}_{p_{0} , \rho_{0} } \leq e^{- \lambda\rho where and the constants called $N$ depend only on $d,\delta,p_{0}$, $n$.

Theorems & Definitions (18)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 3.1
  • Remark 3.2
  • Theorem 3.3: Theorem 4.5 Kr_25
  • Corollary 3.4
  • Lemma 3.5
  • Remark 4.1
  • ...and 8 more