On some properties of the fractional derivative of the Brownian local time (Q6572926)

Let be \(l(t,x)\) the local time of the standard Brownian \(W\) motion at \(x\in {\mathbb R}\) up to \(t.\) It is the only function that satisfies\N\N\[\int_0^t f(W_s)ds=\int_{\mathbb R} f(y)l(t,y)dy\] for any continuous and bounded function \(f.\) It is well known that we can always choose a version of \(l(t,x)\) continuous in \(t\) and \(x.\)\N\NThe fractional derivative of order \(\alpha\in [0,\frac{1}{2})\) of a function \(\psi\) is defined as\N\[\ND_{\alpha}\psi(x):=\int_{\mathbb R} \frac{\psi(x-y)-\psi(x)}{|y|^{1+\alpha}} dy,\N\]\Nwhere \(\psi\) is assumed to be infinitely differentiable and with compact support. But the formula is valid for a class of more general functions including Hölder continuous functions with exponent greater than \(\alpha\) and this includes local time that is Hölder continuous with any exponent less than \(\frac{1}{2}.\) Then we have\N\[\ND_{\alpha}l(t,x):=\int_{\mathbb R} \frac{l(t,x-y)-l(t,x)}{|y|^{1+\alpha}} dy.\N\]\NMoreover, this fractional derivative is continuous in \(t\) and \(x\) with probability 1.\N\NIt is well-known that \(l(t,x)\) can be represented as\N\[\Nl(t,x)=\int_0^t \delta(x-W_s)ds,\N\]\Nwhere \(\delta\) is the Dirac generalized function at \(0.\)\N\NThe first result of the paper is a similar representation for \(D_{\alpha}l(t,x)\) in terms of the generalized function\N\[\Nq_{\alpha}(x):=\frac{1}{|x|^{1+\alpha}}.\N\]\NConcretely,\N\[\ND_{\alpha}l(t,x)=\int_0^t q_{\alpha}(x-W_s)ds=\int_0^t \frac{1}{|x-W_s|^{1+\alpha}}ds.\N\]\NThis object appears in the Itô formula for \(|x-W_t|^{1-\alpha}\) as it is shown in the paper.\N\NThis result is similar, but not derived from, of the Itô formula for\N\[\N|W_t|^{1-\alpha} \mathrm{sgn}(x-W_t),\N\]\Ngiven in [\textit{A. S. Cherny}, Lect. Notes Math. 1755, 348--370 (2001; Zbl 0977.60075)], where the function\N\[\N{\tilde q}_{\alpha}(x)=\frac{\mathrm{sgn}(x)}{|x|^{1+\alpha}}\N\]\Nappears instead of \(q_{\alpha}.\)\N\NFinally, the paper shows that this object also appears in the Feynman-Kac formula where the potential is \(q_{\alpha}.\) This allows to show interesting limit results related with \(D_{\alpha}l(t,x).\)