Multidimensional Ultrametric Pseudodifferential Equations

S. Albeverio ; S. V. Kozyrev

Informatics and Automation, Selected topics of mathematical physics and $p$-adic analysis, Tome 265 (2009), pp. 19-35.

Résumé

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and the space $D'_0(X)$ of generalized functions on a multidimensional ultrametric space. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. We prove an existence theorem and describe all solutions for the Cauchy problem (an analog of the Kovalevskaya theorem).

Bibliographie

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

[2] Khrennikov A. Yu., Nilsson M., $p$-Adic deterministic and random dynamics, Kluwer, Dordrecht, 2004 | MR | Zbl

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

[4] Kozyrev S. V., Metody i prilozheniya ultrametricheskogo i $p$-adicheskogo analiza: ot teorii vspleskov do biofiziki, Sovr. probl. matematiki, 12, MIAN, M., 2008 ; http://www.mi.ras.ru/spm/pdf/012.pdf | DOI

[5] Albeverio S., Karwowski W., “A random walk on $p$-adics – the generator and its spectrum”, Stoch. Processes and Appl., 53\:1 (1994), 1–22 | DOI | MR | Zbl

[6] Albeverio S., Zhao X., “On the relation between different constructions of random walks on $p$-adics”, Markov Processes and Relat. Fields, 6 (2000), 239–255 | MR | Zbl

[7] Yasuda K., “Additive processes on local fields”, J. Math. Sci. Univ. Tokyo, 3 (1996), 629–654 | MR | Zbl

[8] Evans S., “Local fields, Gaussian measures, and Brownian motions”, Topics in probability and Lie groups: boundary theory, CRM Proc. Lect. Notes, 28, Amer. Math. Soc., Providence, RI\, 2001, 11–50 | MR | Zbl

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

[10] Benedetto J. J., Benedetto R. L., “A wavelet theory for local fields and related groups”, J. Geom. Anal., 14:3 (2004), 423–456 | DOI | MR | Zbl

[11] Kozyrev S. V., Khrennikov A. Yu., “Psevdodifferentsialnye operatory na ultrametricheskikh prostranstvakh i ultrametricheskie vspleski”, Izv. RAN. Ser. mat., 69:5 (2005), 133–148 | DOI | MR | Zbl

[12] Khrennikov A. Yu., Kozyrev S. V., “Wavelets on ultrametric spaces”, Appl. and Comput. Harmon. Anal., 19 (2005), 61–76 | DOI | MR | Zbl

[13] Kozyrev S. V., “Vspleski i spektralnyi analiz ultrametricheskikh psevdodifferentsialnykh operatorov”, Mat. sb., 198:1 (2007), 103–126 | DOI | MR | Zbl

[14] Shelkovich V. M., Skopina M., $p$-Adic Haar multiresolution analysis, E-print , 2007 arxiv: 0704.0736 | Zbl

[15] Albeverio S., Kozyrev S. V., Coincidence of the continuous and discrete $p$-adic wavelet transforms, E-print , 2007 arxiv: math-ph/0702010

[16] Kozyrev S. V., “$p$-Adicheskie psevdodifferentsialnye operatory i $p$-adicheskie vspleski”, TMF, 138:3 (2004), 383–394 | DOI | MR | Zbl

[17] Albeverio S., Khrennikov A. Yu., Shelkovich V. M., “Harmonic analysis in the $p$-adic Lizorkin spaces: fractional operators, pseudo-differential equations, $p$-adic wavelets, Tauberian theorems”, J. Fourier Anal. and Appl., 12:4 (2006), 393–425 | DOI | MR | Zbl

[18] Schaefer H. H., Topological vector spaces, Springer, New York, 1999 | MR