A limit theorem for the $L^p$-modulus of continuity of Brownian local time

Simon Campese

Published 2015-06-24Version 1

We proof a limit theorem for the $L^p$-modulus of continuity of Brownian local time. As special cases for $p=2$ and $p=3$, we obtain previous results by Chen et al. (Ann. Prob. 38, 2010, no. 1) and Rosen~(Stoch. Dyn. 11, 2011, no. 1), which were later reproven by Hu and Nualart~(Electron. Commun. Probab. 14, 2009; Electron. Commun. Probab. 15, 2010) and Rosen~(S\'eminaire de Probabilit\'es XLIII, Springer, 2011). In comparison to the previous methods of proof, we follow a fundamentally different approach by exclusively working in the space variable of the Brownian local time, which allows to give a unified argument for arbitrary integer power $p$. The main ingredients are Perkins' semimartingale decomposition, the Kailath-Segall identity and an asymptotic Ray-Knight Theorem by Pitman and Yor.