Self-organized regime switching in null-recurrent dynamics

Based on discrete observations $X_0,X_Delta,dots, X_{nDelta}$ for $Delta=n^{-gamma}$ with $gammain [0,1)$ of the null-recurrent dynamic $dX_t = sigma(X_t)dW_t$ with a Brownian motion $W$ and $sigma(x)=alphamathbb{1}{x