Estimates for the distances between solutions to Kolmogorov equations with diffusion matrices of low regularity

TL;DR

This paper provides new estimates for the difference between solutions to Kolmogorov equations with low-regularity diffusion matrices, avoiding the need for Sobolev derivatives, under Dini mean oscillation conditions.

Contribution

It introduces estimates that do not rely on Sobolev derivatives, applicable to diffusion matrices with low regularity satisfying Dini mean oscillation.

Findings

Estimates for solution differences in weighted L^1 norm.

Applicable to non-singular, bounded diffusion matrices with Dini mean oscillation.

No Sobolev derivatives required in the estimates.

Abstract

We obtain estimates for the weighted $L^1$-norm of the difference of two probability solutions to Kolmogorov equations in terms of the difference of the diffusion matrices and the drifts. Unlike the previously known results, our estimate does not involve Sobolev derivatives of solutions and coefficients. The diffusion matrices are supposed to be non-singular, bounded and satisfy the Dini mean oscillation condition.