$L_1$ and $L_{\infty}$ stability of transition densities of perturbed diffusions

TL;DR

This paper establishes $L_1$ and $L_{}$ stability results for transition densities of perturbed diffusions under weak conditions, allowing unbounded drifts and using a novel convergence metric.

Contribution

It introduces a new stability analysis for diffusions with unbounded drifts, using a weaker convergence condition and a specialized parametrix expansion.

Findings

Derived $L_1$ and $L_{}$ stability bounds for transition densities.

Allowed unbounded drifts with linear growth in the analysis.

Introduced a new convergence metric related to the Holder norm.

Abstract

In this paper, we derive a stability result for $L_1$ and $L_{\infty}$ perturbations of diffusions under weak regularity conditions on the coefficients. In particular, the drift terms we consider can be unbounded with at most linear growth, and we do not require uniform convergence of perturbed diffusions. Instead, we require a weaker convergence condition in a special metric introduced in this paper, related to the Holder norm of the diffusion matrix differences. Our approach is based on a special version of the McKean-Singer parametrix expansion.