SOME LIMIT THEOREMS CONNECTED WITH BROWNIAN LOCAL TIME

Let B = (Bt)t≥0 be a standard Brownian motion and let (Lt ; 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 Lt . We give an illustrative example of the nonlinearity of the integration with respect to local time in the random case.

Here we are, more generally, interested in the limit in L 1 : 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 x t ; 0 ≤ t ≤ 1, x ∈ R) does not admit a linear extension in the random case (see Section 3.2 for details) and in particular local time is not a 1-integrator, which is also proved by Eisenbaum [2].

Notation and preliminaries
Let B = (B t ) t≥0 be a standard Brownian motion and let (L x t ; t ≥ 0, x ∈ R) be a continuous version of its local time process. Let (Ᏺ t ) t≥0 denote the natural filtration generated by B. Without loss of generality, we restrict our attention to functions defined on [0,1] × R.
For a measurable function f from [0,1] × R into R, define the norm · by Let Ᏼ be the set of functions f such that f < ∞. In Eisenbaum [2], it is shown that the integration with respect to L is possible in the following sense. Let f Δ be an elementary function on [0,1] × R, meaning that For such a function, integration with respect to L is defined by Let f be an element of Ᏼ. For any sequence of elementary functions ( f Δk ) k∈N converging to f in Ᏼ, the sequence ( 1 0 R f Δk (s,x)dL x s ) k∈N converges in L 1 . The limit obtained does not depend on the choice of the sequence ( f Δk ) and represents the integral [2]). Let (A(x,t); x ∈ R, 0 ≤ t ≤ 1) be a continuous random process taking values in R, such that for any t in [0,1] and any ω, A(·,t) is absolutely continuous with respect to dx. Note ∂A/∂x its derivative and ask ∂A/∂x to be continuous. Then s exists and the following hold: (2.6) Raouf Ghomrasni 3

Deterministic case
Theorem 3.1. Let F be a bounded element of Ᏼ. The following equalities hold in L 1 : (1) If we take F(t,x) = 1 (x≤a) in (3.1), we have the very definition of L a t .

Proof. We apply Theorem 3.1 to the function F(t,x) = g(t)I(x < b(t)). It follows that
We conclude using (see [4,Corollary 2.9]) that for the continuous function g, we have

Random function case.
Let a,b be in R with a < b. Let ᏹ be the set of elementary processes A such that where (s i ) 1≤i≤n is a subdivision of (0,1], (x j ) 1≤ j≤m is a finite sequence of real numbers in (a,b], Δ = {(s i ,x j ),1 ≤ i ≤ n,1 ≤ j ≤ m}, and, is A i j an Ᏺ sj -measurable random variable such that |A i j | ≤ 1 for every (i, j).

Some limit theorems connected with Brownian local time
Eisenbaum [2] asked the following question: does integration with respect to (L x t ; 0 ≤ t ≤ 1, x ∈ R) admit a linear extension to ᏼ the field generated by ᏹ, verifying the following property?
If (A n ) n≥0 converges a.e. to A(t,x), then ( Consequently, integration with respect to (L x t ; 0 ≤ t ≤ 1, x ∈ R) does not admit a continuous extension in L 1 .
Here we give an illustrative example, thanks to a result obtained by Walsh, which shows the lack of a linear extension.
Let us define A ε (t, We see easily that A ε (t,x) (resp., A ε (t,x)) converges a.e. to L x t , nevertheless we have The limits exist in probability, uniformly for t in compact sets.
Our example follows by recalling the following property: