Geodesic
$p$-Adic Pseudodifferential Operators: Methods and Applications
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 154-165.

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Various methods of the theory of $p$-adic pseudodifferential operators are discussed, such as constructing a basis of $p$-adic wavelets and constructing a large class of $p$-adic pseudodifferential operators that are diagonal in the basis of $p$-adic wavelets but cannot be diagonalized by the Fourier transform. The application of the methods of the theory of $p$-adic pseudodifferential operators to describing replica symmetry breaking in the replica method of the spin-glass theory is demonstrated. 
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S. V. Kozyrev. $p$-Adic Pseudodifferential Operators: Methods and Applications. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Selected topics of $p$-adic mathematical physics and analysis, Tome 245 (2004), pp. 154-165. http://geodesic.mathdoc.fr/item/TM_2004_245_a15/

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

[2] Vladimirov V. S., “O spektrakh nekotorykh psevdodifferentsialnykh operatorov nad polem $p$-adicheskikh chisel”, Algebra i analiz, 2:6 (1990), 107–124 | MR

[3] Kochubei A. N., “Additive and multiplicative fractional differentiations over the field of $p$-adic numbers”, $p$-Adic functional analysis, Lect. Notes Pure and Appl. Math., 192, M. Dekker, New York, 1997, 275–280 | MR | Zbl

[4] Vladimirov V. S., “O razvetvlennykh kharakterakh gruppy idelei odnoklassnykh kvadratichnykh polei”, Tr. MIAN, 224, 1999, 122–129 | MR | Zbl

[5] Kozyrev S. V., “Analiz vspleskov kak $p$-adicheskii spektralnyi analiz”, Izv. RAN. Ser. mat., 66:2 (2002), 149–158 ; arXiv: /math-ph/0012019 | MR | Zbl

[6] Kozyrev S. V., “$p$-Adicheskie psevdodifferentsialnye operatory i $p$-adicheskie vspleski”, Teor. i mat. fizika, 138:3 (2004), 383–394 ; arXiv: /math-ph/0303045 | MR | Zbl

[7] Avetisov V. A., Bikulov A. H., Kozyrev S. V., “Application of $p$-adic analysis to models of breaking of replica symmetry”, J. Phys. A: Math. and Gen., 32:50 (1999), 8785–8791 ; arXiv: /cond-mat/9904360 | DOI | MR | Zbl

[8] Parisi G., Sourlas N., “$p$-Adic numbers and replica symmetry breaking”, Eur. Phys. J. B., 14:3 (2000), 535–542 ; arXiv: /cond-mat/9906095 | DOI | MR

[9] Avetisov V. A., Bikulov A. H., Kozyrev S. V., Osipov V. A., “$p$-Adic models of ultrametric diffusion constrained by hierarchical energy landscapes”, J. Phys. A: Math. and Gen., 35:2 (2002), 177–189 ; arXiv: /cond-mat/0106506 | DOI | MR | Zbl

[10] Avetisov V. A., Bikulov A. Kh., Osipov V. A., “$p$-Adic description of characteristic relaxation in complex systems”, J. Phys. A: Math. and Gen., 36:15 (2003), 4239–4246 ; arXiv: /cond-mat/0210447 | DOI | MR | Zbl

[11] Kochubei A. N., Pseudo-differential equations and stochastics over non-Archimedean fields, M. Dekker, New York, 2001 | MR | Zbl

[12] Khrennikov A., $p$-Adic valued distributions in mathematical physics, Kluwer, Dordrecht, 1994 | MR | Zbl

[13] Khrennikov A., Non-Archimedean analysis: Quantum paradoxes, dynamical systems and biological models, Kluwer, Dordrecht, 1997 | MR | Zbl

[14] Volovich I. V., “$p$-Adic string”, Class. and Quant. Grav., 4 (1987), L83–L87 | DOI | MR

[15] Brekke L., Freund P. G. O., Olson M., Witten E., “Non-archimedean string dynamics”, Nucl. Phys. B., 302 (1988), 365–402 | DOI | MR

[16] Vladimirov V. S., Volovich I. V., “$p$-Adic quantum mechanics”, Commun. Math. Phys., 123 (1989), 659–676 | DOI | MR | Zbl

[17] Djordjevic G. S., Dragovich B., “$p$-Adic path integrals for quadratic actions”, Mod. Phys. Lett. A., 12:20 (1997), 1455–1464 | DOI | MR

[18] Aref'eva I. Ya., Dragovic B., Frampton P., Volovich I. V., “Wave function of the universe and $p$-adic gravity”, Mod. Phys. Lett. A., 6 (1991), 4341–4358 | MR

[19] Albeverio S., Karwowski W., “A random walk on $p$-adic numbers”, Stochastic processes, physics and geometry, II: Proc. III Intern. Conf. (Locarno, 1991), eds. S. Albeverio, U. Cattaneo, D. Merlini, World Sci., Singapore, 1995, 61–74 | MR | Zbl

[20] Kochubei A. N., “Parabolicheskie uravneniya nad polem $p$-adicheskikh chisel”, Izv. AN SSSR. Ser. mat., 55:6 (1991), 1312–1330

[21] Ogielski A. T., Stein D. L., “Dynamics on ultrametric spaces”, Phys. Rev. Lett., 55:15 (1985), 1634–1637 | DOI | MR

[22] Yoshino H., Hierarchical diffusion, aging and multifractality, , 1996 arXiv: /cond-mat/9604033 | MR

[23] Mezard M., Parisi G., Virasoro M., Spin-glass theory and beyond, World Sci., Singapore, 1987 | MR | Zbl

[24] Carlucci D. M., De Dominicis C., “On the replica Fourier transform”, C. R. Acad. Sci. Paris. Ser. IIB: Mech. Phys. Chem. Astr., 325 (1997), 527–530 ; arXiv: /cond-mat/9709200 | Zbl

[25] De Dominicis C., Carlucci D. M., Temesvari T., “Replica Fourier transforms on ultrametric trees, and block-diagonalizing multi-replica matrices”, J. Phys. I (France), 7 (1997), 105–115 ; arXiv: /cond-mat/9703132 | DOI | MR