# Harnack and log Harnack Inequalities for $G$-SDEs with Multiplicative Noise

## TL;DR

This paper establishes Harnack and log Harnack inequalities for G-SDEs with multiplicative noise, extending classical results to the nonlinear G-expectation framework and providing gradient estimates.

## Contribution

It introduces new inequalities for G-SDEs with multiplicative noise, extending existing linear expectation results to the nonlinear G-expectation setting.

## Key findings

- Derived Harnack inequalities for G-SDEs with multiplicative noise
- Extended inequalities to the nonlinear G-expectation framework
- Generalized gradient estimates for these equations

## Abstract

The Harnack and log Harnack inequalities for stochastic differential equation driven by $G$-Brownian motion with multiplicative noise are derived by means of coupling by change of mesure. All of the above results extend the existing ones in the linear expectation setting. Moreover, the gradient estimate generalize the nonlinear results appeared in [11].