Absolute continuity of Rosenblatt measures

TL;DR

This paper investigates conditions under which shifted Rosenblatt measures on path space are absolutely continuous, extending previous results by identifying specific Gaussian shifts that preserve measure equivalence.

Contribution

It demonstrates that certain Gaussian shifts, including deterministic and fractional Brownian motion-based components, lead to absolute continuity of Rosenblatt measures.

Findings

Existence of absolutely continuous measures under specific Gaussian shifts.

Extension of previous non-existence results to new classes of shifts.

Examples illustrating the applicability of the main theorem.

Abstract

In the article, we address the problem of absolute continuity of translated Rosenblatt measures on the path space. In [\v{C}oupek, P., K\v{r}\'i\v{z}, P., Maslowski, B., Stoch. Proc. Appl. 179 (2025) art. no. 104499], it is shown that there is no probability measure that would be equivalent to the original probability measure and under which a Rosenblatt path with a linear drift would again be a Rosenblatt path. Here, we show that if the Rosenblatt path is shifted in a direction belonging to a class of nontrivial Gaussian variables (that consists of a deterministic shift and a Wiener integral with respect to a fractional Brownian motion with a related Hurst parameter), such a measure exists. We also give several examples to demonstrate the scope of the result.