Some limit theorems connected with Brownian local time

Abstract

Let B=(Bt)t≥0 be a standard Brownian motion and let (Ltx;t≥0,x∈ℝ) be a continuous version of its local time process. We show that the following limitlim⁡ε↓0(1/2ε)∫0t{F(s,Bs−ε)−F(s,Bs+ε)}ds is well defined for a large class of functions F(t,x), and moreover we connect it with the integration with respect to local time Ltx . We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.