Geodesic
On Some Properties of the Fractional Derivative of the Brownian Local Time
Informatics and Automation, 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. Informatics and Automation, Noncommutative Analysis and Quantum Information Theory, Tome 324 (2024), pp. 109-123. http://geodesic.mathdoc.fr/item/TRSPY_2024_324_a10/

[1] Biane Ph., Yor M., “Valeurs principales associées aux temps locaux browniens”, Bull. sci. math., 111 (1987), 23–101 | MR | Zbl

[2] M. S. Birman and M. Z. Solomyak, Spectral Theory of Self-adjoint Operators in Hilbert Space, Kluwer, Dordrecht, 1987 | MR | Zbl

[3] A. N. Borodin and I. A. Ibragimov, Limit Theorems for Functionals of Random Walks, Proc. Steklov Inst. Math., 195, Am. Math. Soc., Providence, RI, 1995 | MR | Zbl

[4] A. N. Borodin and P. Salminen, Handbook of Brownian Motion: Facts and Formulae, Birkhäuser, Basel, 2002 | MR | Zbl

[5] Bouleau N., Yor M., “Sur la variation quadratique des temps locaux de certaines semimartingales”, C. r. Acad. sci. Paris. Ser. 1, 292 (1981), 491–494 | MR | Zbl

[6] Cherny A.S., “Principal values of the integral functionals of Brownian motion: Existence, continuity and an extension of Ito's formula”, Séminaire de probabilités XXXV, Lect. Notes Math., 1755, Springer, Berlin, 2001, 348–370 | DOI | MR | Zbl

[7] Föllmer H., Protter Ph., Shiryayev A.N., “Quadratic covariation and an extension of Ito's formula”, Bernoulli, 1:1–2 (1995), 149–169 | DOI | MR | Zbl

[8] I. M. Gel'fand and G. E. Shilov, Generalized Functions, v. 1, Properties and Operations, Academic Press, New York, 1964 | MR

[9] Naimark M.A., Lineinye differentsialnye operatory, Nauka, M., 1969

[10] A. M. Savchuk and A. A. Shkalikov, “Sturm–Liouville operators with singular potentials”, Math. Notes, 66:6 (1999), 741–753 | DOI | DOI | MR | Zbl