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On the fractional derivative of Brownian local times. (English) Zbl 0625.60090
Let $$L^ a_ t$$ denote a jointly continuous version of the local time of Brownian motion $$B_ t$$. Let $$0<\alpha <$$, and define a function $$F_ a(x)$$ to be 0 for $$x<a$$, and $(x-a)^{1-\alpha}/[(1-\alpha)(- \alpha)]\quad for\quad x\geq \alpha.$ Theorem: If $$\beta\in (\alpha,1]$$, and if f is Hölder continuous of order $$\beta$$, then $\int^{t}_{0}(D^{\alpha}g)(B_ s)ds=\Gamma (-\alpha)^{- 1}\int^{\infty}_{-\infty}H^ a(-1-\alpha,t)g(a)da$ $where\quad H^ a(-1-\alpha,t)=2F_ a(B_ t)-2F_ a(B_ 0)-2\int^{t}_{0}F_ a'(B_ s)dB_ s.$ This problem is used to obtain the main result of this paper, which is a representation of the fractional derivative $$D^{\alpha}L^._ t$$ with respect to the space co-ordinate, in terms of the function H and its Hilbert transform.
Reviewer: R.W.R.Darling

##### MSC:
 60J55 Local time and additive functionals 60J65 Brownian motion
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