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


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