Some limit theorems connected with Brownian local time.

Publication: Journal of Applied Mathematics and Stochastic Analysis

Publication Date: 01-JAN-06

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COPYRIGHT 2006 Hindawi Publishing Corp.

Let B = [([B.sub.t]).sub.t[greater than or equal to]0] be a standard Brownian motion and let ([L.sup.x.sub.t] ; t [greater than or equal to] 0, x [member of] R) be a continuous version of its local time process. We show that the following limit [lim.sub.[epsilon][down arrow]0](1/2[epsilon]) [[integral].sup.t.sub.0] {F(s, [B.sub.s] - [epsilon]) - F(s, [B.sub.s] + [epsilon])}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 [L.sup.x.sub.t]. We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.

1. Introduction

1.1. The local time of the Brownian motion B at the point a is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.1)

which equivalently could be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.2)

Here we are, more generally, interested in the limit in [L.sup.1]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

for some function F.

Our motivation comes from the desire to connect Chitashvili and Mania results [1] with those of Eisenbaum [2].

1.2. We give an example which illustrates that the integration with respect to ([L.sup.x.sub.t] ; [less than or equal to] t [less than or equal to] 1, x [member of] R) does not admit a linear extension in the random case (see Section 3.2 for details)...

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