On Non-degenerate Chaos Processes

TL;DR

This paper investigates non-degenerate properties of chaos processes within Wiener chaos and demonstrates that solutions to SDEs driven by such processes possess densities, using Malliavin calculus techniques.

Contribution

It establishes non-degenerate properties for chaos processes and proves SDE solutions driven by these processes have densities, advancing understanding of their probabilistic structure.

Findings

Chaos processes in Wiener chaos are non-degenerate under certain conditions

Solutions to SDEs driven by these processes admit smooth densities

The approach combines Malliavin calculus with Wiener space analysis

Abstract

We consider a process $\{X_t\}_{0\leq t\leq 1}$ in a fixed Wiener chaos $\mathcal{H}_n$. We establish some non-degenerate properties and related results for $\{X_t\}_{0\leq t\leq 1}$. As an application, we show that solution to SDE driven by $\{X_t\}_{0\leq t\leq 1}$ admits a density. Our approach relies on an interplay between Malliavin calculus and analysis on Wiener space.