Hurst exponent analysis is a study of the long-memory capacity of the time series in question. If the time series is not stationary, it must be transformed into a stationary series before this analysis.

The time series must not be stationary but the increments of the time series must be stationary. If you do not observe stationarity of the increments, you may estimate a time-varying Hurst exponent (which defines a multifractional Brownian motion). I join a paper to explain how to deal with this. As you can see in equation 3 of the paper, the log of any moment of the increments linearly depends on H(t), which is the time-varying Hurst exponent.

Article Estimation of time-dependent Hurst exponents with variationa...