Geodesic
On Some Properties of the Fractional Derivative of the Brownian Local Time
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 109-123.

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We study the properties of the fractional derivative $D_\alpha l(t,x)$ of order $\alpha 1/2$ of the Brownian local time $l(t,x)$ with respect to the variable $x$. This derivative is understood as the convolution of the local time with the generalized function $|x|^{-1-\alpha }$. We show that $D_\alpha l(t,x)$ appears naturally in Itô's formula for the process $|w(t)|^{1-\alpha }$. Using the martingale technique, we also study the limit behavior of $D_\alpha l(t,x)$ as $t\to \infty $.
Keywords: stochastic processes, local time, fractional derivative. 
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I. A. Ibragimov; N. V. Smorodina; M. M. Faddeev. On Some Properties of the Fractional Derivative of the Brownian Local Time. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 109-123. http://geodesic.mathdoc.fr/item/TM_2024_324_a10/

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