Geodesic
Feynman Formulas and Path Integrals for Evolution Equations with the Vladimirov Operator
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 229-240.

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We obtain Feynman formulas in the momentum space and Feynman–Kac formulas in the momentum and phase spaces for a $\mathfrak p$-adic analog of the heat equation in which the role of the Laplace operator is played by the Vladimirov operator. We also present the Feynman and Feynman–Kac formulas in the configuration space that have been proved in our previous papers under additional constraints. In all these formulas, integration is performed with respect to countably additive measures. The technique developed in the paper is fundamentally different from that used by the authors when studying path integrals in configuration spaces. In particular, the paper extensively uses the infinite-dimensional Fourier transform. 
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O. G. Smolyanov; N. N. Shamarov. Feynman Formulas and Path Integrals for Evolution Equations with the Vladimirov Operator. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 229-240. http://geodesic.mathdoc.fr/item/TM_2009_265_a19/

[1] Smolyanov O. G., fon Vaitszekker Kh., Vittikh O., “Integraly po poverkhnostyam v rimanovom prostranstve i formuly Feinmana”, DAN, 408:6 (2006), 738–742 | MR

[2] Smolyanov O. G., Shavgulidze E. T., Kontinualnye integraly, Izd-vo MGU, M., 1990. | MR | Zbl

[3] Vladimirov V. S., Volovich I. V., Zelenov E. I., $p$-Adicheskii analiz i matematicheskaya fizika, Nauka, M., 1994 | MR

[4] Smolyanov O. G., Shamarov N. N., “Predstavleniya funktsionalnymi integralami reshenii uravneniya teploprovodnosti s operatorom Vladimirova”, Vestn. Mosk. un-ta. Matematika. Mekhanika, 2008, no. 4, 16–22

[5] Davies E. B., One-parameter semigroups, Acad. Press, London, 1980 | MR | Zbl

[6] Avetisov V. A., Bikulov A. Kh., Osipov V. A., “$p$-Adicheskie modeli ultrametricheskoi diffuzii v konformatsionnoi dinamike makromolekul”, Tr. MIAN, 245, 2004, 55–64 | MR | Zbl

[7] Kozyrev S. V., Metody i prilozheniya ultrametricheskogo i $p$-adicheskogo analiza: ot teorii vspleskov do biofiziki, Sovrem. probl. matematiki, 12, MIAN, M., 2008 | DOI

[8] Chernoff P. R., “Note on product formulas for operator semigroups”, J. Funct. Anal., 2:2 (1968), 238–242 | DOI | MR | Zbl

[9] Shamarov N. N., “Mera Puassona–Maslova i formuly Feinmana dlya resheniya uravneniya Diraka”, Fund. i prikl. matematika, 12:6 (2006), 193–211 | MR

[10] Shamarov N. N., “Matrix-valued cylindrical measures of Markov type and their Fourier transforms”, Russ. J. Math. Phys., 10:3 (2003), 319–333 | MR | Zbl

[11] Maslov V. P., Kompleksnye markovskie tsepi i kontinualnyi integral Feinmana dlya nelineinykh uravnenii, Nauka, M., 1976 | MR

[12] Shamarov N. N., “Poisson–Maslov type formulas for Schrödinger equations with matrix-valued potentials”, Infin. Dimens. Anal. Quantum Probab. and Relat. Top., 10:4 (2007), 641–649 | DOI | MR | Zbl