Asymptotical stability of differential equations driven by H\"older--continuous paths

In this manuscript, we establish asymptotic local exponential stability of the trivial solution of differential equations driven by H"older–continuous paths with H"older exponent greater than $1\mathrm{/}2$. This applies in particular to stochastic differential equations driven by fractional Brownian motion with Hurst parameter greater than $1\mathrm{/}2$. We motivate the study of local stability by giving a particular example of a scalar equation, where global stability of the trivial solution can be obtained.