SOME LIMIT THEOREMS CONNECTED WITH BROWNIAN LOCAL TIME

Let B = (Bt)t≥0 be a standard Brownian motion and let (Lxt; t ≥ 0, x ∈R) be a continuous version of its local time process. We show that the following limit limε↓0(1/2ε) ∫ t 0{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 Lxt. We give an illustrative ex-ample of the nonlinearity of the integration with respect to local time in the random case. Copyright © 2006 Raouf Ghomrasni. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1