On the fractional derivative of Brownian local times (Q579766)

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.