# Behavior with respect to the Hurst index of the Wiener Hermite integrals and application to SPDEs

## TL;DR

This paper investigates the limit behavior of Wiener Hermite integrals with respect to multi-parameter Hermite processes as their Hurst indices approach critical values, with applications to SPDEs like the stochastic heat equation.

## Contribution

It provides a detailed analysis of the distributional limits of Wiener Hermite integrals with varying Hurst parameters and applies these results to specific stochastic partial differential equations.

## Key findings

- Limit behavior characterized as Hurst indices approach 1 or 1/2.
- Applications to stochastic heat equation with Hermite noise.
- Analysis of Hermite Ornstein-Uhlenbeck process limits.

## Abstract

We consider the Wiener integral with respect to a $d$-parameter Hermite process with Hurst multi-index ${\bf H}= (H_{1},\ldots, H_{d}) \in \left( \frac{1}{2}, 1\right) ^{d}$ and we analyze the limit behavior in distribution of this object when the components of ${\bf H}$ tend to $1$ and/or $\frac{1}{2}$. As examples, we focus on the solution to the stochastic heat equation with additive Hermite noise and to the Hermite Ornstein-Uhlenbeck process.