Joint continuity and representations of additive functionals of d- dimensional Brownian motion.(English)Zbl 0543.60080

A correspondence exists between additive functionals, $$A_ t$$, of d- dimensional Brownian motion and measures, $$\mu$$, on $$R^ d$$. When $$d=1$$, a well-known result of Ito and Mc’Kean states that every $$A_ t$$ can be represented by (1) $$A_ t=\int L(t,y)\mu(dy)$$, where $$\mu$$ is the corresponding measure on R and L(t,y) is the local time at y.
The author first gives conditions on $$\mu_ a$$, 0$$\leq a\leq 1$$, which guarantee that the corresponding $$A^ a_ t$$ will be jointly continuous in a and t almost surely. This result is then used to produce a d- dimensional analogue to (1).
For $$d>1$$ no local time exists. However, if $$W_ t=(W^ 1_ t$$, $$W^ 2_ t,...,W^ d_ t)$$ represents a d-dimensional Brownian motion and $$\cdot$$ denotes inner product, we can let L(t,s,v) denote the local time of $$W_ t\cdot v$$ at s. If $$B=\{v:| v| =1\}$$, set $$A^ b_ t=\int \int_{B}\int^{\infty}_{-\infty}I_ b(s-y\cdot v)\quad L(t,s,v)\quad ds\quad dv\quad d\mu$$, where $$I_ b(y)=(2\pi)^{- d}\int_{0}^{\infty}\cos(qy)q^{d-1}\exp(-bq^ 2/2)dq$$. The following result, analogous to (1), is shown. If $$\mu$$ is any measure such that $$\mu(\{y:| y-x|<\delta \})\leq c\delta^{d-2+\nu}$$ for some constants $$c,\nu>0$$ independent of x, then for each $$u>0$$ we have, almost surely, $$\lim_{b\to 0} \sup_{t\leq u}| A_ t-A^ b_ t| =0$$.
Reviewer: E.Boylan

MSC:

 60J55 Local time and additive functionals 60H05 Stochastic integrals 60G15 Gaussian processes
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References:

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