Logarithmic stability estimates for initial data in Ornstein-Uhlenbeck equation on $L^2$-space

TL;DR

This paper establishes new logarithmic stability estimates for initial data in the Ornstein-Uhlenbeck equation on L^2 space, linking observability and inverse problems for parabolic equations with unbounded coefficients.

Contribution

It extends previous results by providing stability estimates for the non-analytic Ornstein-Uhlenbeck semigroup on L^2 space with Lebesgue measure.

Findings

Proved logarithmic stability estimates for initial data.

Connected observability with inverse problems for unbounded coefficient equations.

Completed the understanding of Ornstein-Uhlenbeck semigroup stability on L^2.

Abstract

In this paper, we continue the investigation on the connection between observability and inverse problems for a class of parabolic equations with unbounded first order coefficients. We prove new logarithmic stability estimates for a class of initial data in the Ornstein-Uhlenbeck equation posed on $L^2\left(\mathbb{R}^N\right)$ with respect to the Lebesgue measure. The proofs combine observability and logarithmic convexity results that include a non-analytic semigroup case. This completes the picture of the recent results obtained for the analytic Ornstein-Uhlenbeck semigroup on $L^2$-space with invariant measure.