Abstract
In the 1980s I. Volovich proposed that the world geometry in regimes smaller than the Planck scale might be non-Archimedean [112, 113]. This hypothesis conducts naturally to consider models involving geometry and analysis over \(\mathbb{Q}_{p}\). Since then, a big number of articles have appeared exploring these and related themes, see e.g. [36], [107, Chapter 6] and the references therein.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Albeverio, S., Belopolskaya, Y.: Stochastic processes in \(\mathbb{Q}_{p}\) associated with systems of nonlinear PIDEs. p-Adic Numbers Ultrametric Anal. Appl. 1 (2), 105–117 (2009)
Albeverio, S., Karwowski, W.: Diffusion in p-adic numbers. In: Ito, K., Hida, H. (eds.) Gaussian Random Fields, pp. 86–99. World Scientific, Singapore (1991)
Albeverio, S., Karwowski, W.: A random walk on p-adics: the generator and its spectrum. Stoch. Process. Appl. 53, 1–22 (1994)
Albeverio, S., Karwowski, W.: Jump processes on leaves of multibranching trees. J. Math. Phys. 49 (9), 093503, 20 pp. (2008)
Albeverio, S., Khrennikov, A.Yu., Shelkovich, V.M.: Theory of p-Adic Distributions: Linear and Nonlinear Models. Cambridge University Press, Cambridge (2010)
Ansari, A., Berendzen, J., Bowne, S.F., Frauenfelder, H., Iben, I.E.T., Sauke, T.B., Shyamsunder, E., Young, R.D.: Protein states and proteinquakes. Proc. Natl. Acad. Sci. USA 82, 5000–5004 (1985)
Arendt, W., Batty, C.J.K., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser-Springer, Basel (2011)
Avetisov, V.A., Bikulov, A.H., Kozyrev, S.V.: Application of p-adic analysis to models of breaking of replica symmetry. J. Phys. A 32 (50), 8785–8791 (1999)
Avetisov, V.A., Bikulov, A.Kh., Kozyrev, S.V.: Description of logarithmic relaxation by a model of a hierarchical random walk. Dokl. Akad. Nauk 368 (2), 164–167 (1999, in Russian)
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 35 (2), 177–189 (2002)
Avetisov, V.A., Bikulov, A.Kh., Osipov, V.A.: p-Adic description of characteristic relaxation in complex systems. J. Phys. A 36 (15), 4239–4246 (2003)
Avetisov, V.A., Bikulov, A.Kh., Osipov, V.A.: p-Adic models of ultrametric diffusion in the conformational dynamics of macromolecules. Proc. Steklov Inst. Math. 245 2, 48–57 (2004)
Avetisov, V.A., Bikulov, A.Kh.: On the ultrametricity of the fluctuation dynamic mobility of protein molecules. Proc. Steklov Inst. Math. 265 (1), 75–81 (2009)
Avetisov, V.A., Bikulov, A.Kh., Zubarev, A.P.: First passage time distribution and the number of returns for ultrametric random walks. J. Phys. A 42 (8), 085003, 18 pp. (2009)
Atiyah, M.F.: Resolution of singularities and division of distributions. Commun. Pure Appl. Math. 23,145–150 (1970)
Bass, R.F., Levin, D.A.: Transition probabilities for symmetric jump processes. Trans. Am. Math. Soc. 354 (7), 2933–2953 (2002)
Berg, C., Forst, G.: Potential Theory on Locally Compact Abelian Groups. Springer, Berlin (1975)
Bernstein, I.N.: Modules over the ring of differential operators; the study of fundamental solutions of equations with constant coefficients. Funct. Anal. Appl. 5 (2), 1–16 (1972)
Beloshapka, O.: Feynman formulas for an infinite-dimensional p-adic heat type equation. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14 (1), 137–148 (2011)
Blair, A.D.: Adelic path integrals. Rev. Math. Phys. 7, 21–49 (1995)
Bikulov, A.Kh., Volovich, I.V.: p-Adic Brownian motion. Izv. Math. 61 (3), 537–552 (1997)
Borevich, Z.I., Shafarevich, I.R.: Number Theory. Academic, London (1986)
Casas-Sánchez, O.F., Zúñiga-Galindo, W.A.: p-Adic elliptic quadratic forms, parabolic-type pseudodifferential equations with variable coefficients and Markov processes. p-Adic Numbers Ultrametric Anal. Appl. 6 (1), 1–20 (2014)
Cazenave, T., Haraux, A.: An Introduction to Semilinear Evolution Equations. Oxford University Press, Oxford (1998)
Chacón-Cortes, L.F., Zúñiga-Galindo, W.A.: Nonlocal operators, parabolic-type equations, and ultrametric random walks. J. Math. Phys. 54, 113503 (2013) [Erratum 55, 109901 (2014)]
Chacón-Cortes, L., Zúñiga-Galindo, W.A.: Non-local operators, non-Archimedean parabolic-type equations with variable coefficients and Markov processes. Publ. Res. Inst. Math. Sci. 51 (2), 289–317 (2015)
Chacón-Cortes, L., Zúñiga-Galindo, W.A.: Heat traces and spectral zeta functions for p-adic laplacians. Accepted in Algebra i Analiz. arXiv:1511.02146
Chen, Z.-Q., Kumagai, T.: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140 (1–2), 277–317 (2008)
Connes, A.: Trace formula in noncommutative geometry and the zeros of the Riemann zeta function. Sel. Math. (N.S.) 5, 29–106 (1999)
Denef, J.: Report on Igusa’s local zeta function. Séminaire Bourbaki 43, exp. 741 (1990–1991); Astérisque 201–202–203, 359–386 (1991). Available at http://www.wis.kuleuven.ac.be/algebra/denef.html
de Jager, E.M.: The Lorentz-invariant solutions of the Klein-Gordon equation. SIAM J. Appl. Math. 15, 944–963 (1967)
de Jager, E.M.: Applications of Distributions in Mathematical Physics. Mathematical Centre Tracts, vol. 10. Mathematisch Centrum, Amsterdam (1964)
Diamond, H.: Elementary methods in the study of the distribution of prime numbers. Bull. Am. Math. Soc. (N.S.) 7 (3), 553–589 (1982)
Dimock, J.: Quantum Mechanics and Quantum Field Theory: A Mathematical Primer. Cambridge University Press, Cambridge (2011)
Dragovich, B.: p-Adic and adelic quantum mechanics. Proc. Steklov Inst. Math. 245 (2), 64–77 (2004)
Dragovich, B., Khrennikov, A.Yu., Kozyrev, S.V., Volovich, I.V.: On p-adic mathematical physics. p-Adic Numbers Ultrametric Anal. Appl. 1 (1), 1–17 (2009)
Dragovich, B., Radyno, Y., Khrennikov, A.: Generalized functions on adeles. J. Math. Sci. (N.Y.) 142 (3), 2105–2112 (2007)
Droniou, J., Gallouet, T., Vovelle, J.: Global solution and smoothing effect for a non-local regularization of a hyperbolic equation. J. Evol. Equ. 3 (3), 499–521 (2003)
Dynkin, E.B.: Markov Processes, vol. I. Springer, Berlin (1965)
Ehrenpreis, L.: Solution of some problems of division. Part I. Division by a polynomial of derivation. Am. J. Math. 76, 883–903 (1954)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Evans, S.N.: Local properties of Lévy processes on a totally disconnected group. J. Theor. Probab. 2 (2), 209–259 (1989)
Evans, S.N.: Local field Brownian motion. J. Theor. Probab. 6 (4), 817–850 (1993)
Folland, G.B.: Quantum Field Theory: A Tourist Guide for Mathematicians. American Mathematical Society, Providence (2008)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Frauenfelder, H., McMahon, B.H., Fenimore, P.W.: Myoglobin, the hydrogen atom of biology and paradigm of complexity. Proc. Natl. Acad. Sci. USA 100 (15), 8615–8617 (2003)
Frauenfelder, H., Sligar, S.G., Wolynes, P.G.: The energy landscape and motions of proteins. Science 254, 1598–1603 (1991)
Galeano-Peñaloza, J., Zúñiga-Galindo, W.A.: Pseudo-differential operators with semi-quasielliptic symbols over p-adic fields: J. Math. Anal. Appl. 386 (1), 32–49 (2012)
Gel’fand, I.M., Shilov, G.E.: Generalized Functions, vol. 1. Academic, New York (1977)
Goldfeld, D., Hundley, J.: Automorphic Representations and L-Functions for the General Linear Group, vol. I. Cambridge University Press, Cambridge (2011)
Haran, S.: Potentials and explicit sums in arithmetic. Invent. Math. 101, 797–703 (1990)
Haran, S.: Quantizations and symbolic calculus over the p-adic numbers. Ann. Inst. Fourier 43 (4), 997–1053 (1993)
Harlow, D., Shenker, S., Stanford, D., Susskind, L.: Tree-like structure of eternal inflation: a solvable model. Phys. Rev. D 85 (6) (2012). Article Number: 063516
Hironaka, H.: Resolution of singularities of an algebraic variety over a field of characteristic zero. Ann. Math. 79, 109–326 (1964)
Hörmander, L.: On the division of distributions by polynomials. Ark. Mat. 3, 555–568 (1958)
Hörmander, L.: The Analysis of Linear Partial Differential Operators. II: Differential Operators with Constant Coefficients. Grundlehren der Mathematischen Wissenschaften, vol. 257. Springer, Berlin (1983)
Igusa, J.-I.: An Introduction to the Theory of Local Zeta Functions. AMS/IP Studies in Advanced Mathematics, vol. 14. American Mathematical Society, Providence (2000)
Igusa, J.-I.: Some aspects of the arithmetic theory of polynomials. Discrete Groups in Geometry and Analysis (New Haven, 1984). Progress in Mathematics, vol. 67, pp. 20–47. Birkhäuser, Boston (1987)
Igusa, J.-I.: Zeta distributions associated with some invariants. Am. J. Math. 109 (1), 1–33 (1987)
Jacob, N.: Pseudo Differential Operators and Markov Processes, Vol. II: Generators and Their Potential Theory, xxii+453 pp. Imperial College Press, London (2002)
Karwowski, W.: Diffusion processes with ultrametric jumps. Rep. Math. Phys. 60 (2), 221–235 (2007)
Karwowski, W., Mendes, R.V.: Hierarchical structures and asymmetric stochastic processes on p-adics and adèles. J. Math. Phys. 35 (9), 4637–4650 (1994)
Kigami, J.: Transitions on a noncompact Cantor set and random walks on its defining tree. Ann. Inst. Henri Poincaré Probab. Stat. 49 (4), 1090–1129 (2013)
Khrennikov, A.: p-Adic Valued Distributions in Mathematical Physics. Kluwer, Dordrecht (1994)
Khrennikov, A.: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer, Dordrecht (1997)
Khrennikov, A.Yu., Kozyrev, S.V.: Wavelets on ultrametric spaces. Appl. Comput. Harmon. Anal. 19, 61–76 (2005)
Khrennikov, A.Yu., Kozyrev, S.V.: Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals. Phys. A: Stat. Mech. Appl. 359, 222–240 (2006)
Khrennikov, A.Yu., Kozyrev, S.V.: Replica symmetry breaking related to a general ultrametric space II: RSB solutions and the $n ∖to0$ limit. Phys. A: Stat. Mech. Appl. 359, 241–266 (2006)
Khrennikov, A.Yu., Kozyrev, S.V.: Replica symmetry breaking related to a general ultrametric space III: the case of general measure. Phys. A: Stat. Mech. Appl. 378 (2), 283–298 (2007)
Khrennikov, A.Yu., Kozyrev, S.V.: Ultrametric random field. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2), 199–213 (2006)
Khrennikov, A.Yu., Kozyrev, S.V., Oleschko, K., Jaramillo, A.G., Correa, L.J.: Application of p-adic analysis to time series. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16 (4), 1350030, 15 pp. (2013)
Khrennikov, A.Y., Radyno, Y.V.: On adelic analogue of Laplacian. Proc. Jangjeon Math. Soc. 6 (1), 1–18 (2003)
Khrennikov, A.Y., Shelkovich, V.M., van der Walt, J.H.: Adelic multiresolution analysis, construction of wavelet bases and pseudo-differential operators. J. Fourier Anal. Appl. 19 (6), 1323–1358 (2013)
Khrennikov, A.Yu., Kosyak, A.V., Shelkovich, V.M.: Wavelet analysis on adeles and pseudo-differential operators. J. Fourier Anal. Appl. 18 (6), 1215–1264 (2012)
Khrennikov, A.Yu., Kozyrev, S.V.: Genetic code on the diadic plane. Phys. A: Stat. Mech. Appl. 381, 265–272 (2007)
Khrennikov, A.Yu., Kozyrev, S.V.: 2-Adic clustering of the PAM matrix. J. Theor. Biol. 261, 396–406 (2009)
Kozyrev, S.V., Khrennikov, A.Yu.: Pseudodifferential operators on ultrametric spaces, and ultrametric wavelets. Izv. Math. 69 (5), 989–1003 (2005)
Khrennikov, A.Yu., Kozyrev, S.V.: p-Adic pseudodifferential operators and analytic continuation of replica matrices. Theor. Math. Phys. 144 (2), 1166–1170 (2005)
Kochubei, A.N.: A non-Archimedean wave equation. Pac. J. Math. 235, 245–261 (2008)
Kochubei, A.N.: Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields. Marcel Dekker, New York (2001)
Kochubei, A.N., Parabolic equations over the field of p-adic numbers. Math. USSR Izv. 39, 1263–1280 (1992)
Kozyrev, S.V.: Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics. Proc. Steklov Inst. Math. 274 (1 Suppl.), 1–84 (2011)
Leichtnam, E.: Scaling group flow and Lefschetz trace formula for laminated spaces with p-adic transversal. Bull. Sci. Math. 131 (7), 638–669 (2007)
Malgrange, B.: Existence et approximation des solutions des é quations aux dérivées partielles et des équations de convolution. Ann. Inst. Fourier 6, 271–355 (1955/1956)
Manin, Y.I.: Reflections on Arithmetical Physics. Conformal Invariance and String Theory, pp. 293–303. Academic, New York (1989)
Mézard, M., Parisi, G.: Virasoro Miguel Angel. Spin Glass Theory and Beyond. World Scientific, Singapore (1987)
Ogielski, A.T., Stein, D.L.: Dynamics on ultrametric spaces. Phys. Rev. Lett. 55 (15), 1634–1637 (1985)
Ono, T.: Gauss transforms and zeta-functions. Ann. Math. 91 (2), 332–361 (1970)
Ortner, N., Wagner, P.: A short proof of the Malgrange-Ehrenpreis theorem. In: Dierolf, S., Dineen, S., Domański, P. (eds.) Functional Analysis. Proceedings of the 1st International Workshop in Trier, Germany, 1994, pp. 343–352. de Gruyter, Berlin (1996)
Parisi, G., Sourlas N.: p-Adic numbers and replica symmetry breaking. Eur. Phys. J. B Condens. Matter Phys. 14 (3), 535–542 (2000)
Radyno, Y.V., Radyna, Y.M.: Generalized Functions on Adeles. Linear and Non-linear Theories. Linear and Non-linear Theory of Generalized Functions and Its Applications. Banach Center Publications, vol. 88, pp. 243–250. Polish Academy of Sciences, Institute of Mathematics, Warsaw (2010)
Rallis, S., Schiffmann, G.: Distributions invariantes par le groupe orthogonal. Analyse Harmonique Sur Les Groupes de Lie (Sém., Nancy-Strasbourg, 1973–1975). Lecture Notes in Mathematics, vol. 497, pp. 494–642. Springer, New York (1975)
Ramakrishnan, D., Valenza, R.J.: Fourier Analysis on Number Fields. Springer, New York (1999)
Rammal, R., Toulouse, G., Virasoro, M.A.: Ultrametricity for physicists. Rev. Mod. Phys. 58 (3), 765–788 (1986)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Functional Analysis I. Academic, New York (1980)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. Academic, New York (1975)
Rodríguez-Vega, J.J.: On a general type of p-adic parabolic equations. Rev. Colomb. Mat. 43 (2), 101–114 (2009)
Rodríguez-Vega, J.J., Zúñiga-Galindo, W.A.: Taibleson operators, p-adic parabolic equations and ultrametric diffusion. Pac. J. Math. 237 (2), 327–347 (2008)
Rosay, J.-P.: A very elementary proof of the Malgrange-Ehrenpreis theorem. Am. Math. Mon. 98 (6), 518–523 (1991)
Rudin, W.: Fourier Analysis on Groups. Interscience, New York (1962)
Samko, S.G.: Hypersingular Integrals and Their Applications. Taylor and Francis, London (2002)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives and Some of Their Applications. Gordon and Breach Science Publishers, Yverdon (1993)
Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Row/Peterson, Evanston (1961)
Serre, J.-P.: Lie Algebras and Lie Groups. American Mathematical Society, Providence (1968)
Taibleson, M.H.: Fourier Analysis on Local Fields. Princeton University Press, Princeton (1975)
Torba, S.M., Zúñiga-Galindo, W.A.: Parabolic type equations and Markov stochastic processes on adeles. J. Fourier Anal. Appl. 19 (4), 792–835 (2013)
Varadarajan, V.S.: Reflections on Quanta, Symmetries, and Supersymmetries. Springer, New York (2011)
Varadarajan, V.S.: Path integrals for a class of p-adic Schr ödinger equations. Lett. Math. Phys. 39 (2), 97–106 (1997)
Varadarajan, V.S.: Arithmetic quantum physics: why, what, and whither. Proc. Steklov Inst. Math. 245 (2), 258–265 (2004)
Veys, W., Zúñiga-Galindo, W.A.: Zeta functions for analytic mappings, log-principalization of ideals, and newton polyhedra. Trans. Am. Math. Soc. 360, 2205–2227 (2008)
Vladimirov, V.S., Volovich, I.V., Zelenov, E.I.: p-Adic Analysis and Mathematical Physics. World Scientific, Singapore (1994)
Volovich, I.V.: Number theory as the ultimate physical theory. p-Adic Numbers Ultrametric Anal. Appl. 2 (1), 77–87 (2010)
Volovich, I.V.: p-Adic string. Class. Quantum Grav. 4, L83–L87 (1987)
Wales, D.J., Miller, M.A., Walsh, T.R.: Archetypal energy landscapes. Nature 394, 758–760 (1998)
Weil, A.: Basic Number Theory. Springer, New York (1967)
Yasuda, K.: Markov processes on the adeles and representations of Euler products. J. Theor. Probab. 23 (3), 748–769 (2010)
Zúñiga-Galindo, W.A.: Non-Archimedean white noise, pseudodifferential stochastic equations, and massive Euclidean fields. J. Fourier Anal. Appl. (2016). doi:10.1007/s00041-016-9470-1
Zúñiga-Galindo, W.A.: The non-Archimedean stochastic heat equation driven by Gaussian noise. J. Fourier Anal. Appl. 21 (3), 600–627 (2015)
Zúñiga-Galindo, W.A.: The Cauchy problem for non-Archimedean pseudodifferential equations of Klein-Gordon type. J. Math. Anal. Appl. 420 (2), 1033–1050 (2014)
Zúñiga-Galindo, W.A.: Local zeta functions and fundamental solutions for pseudo-differential operators over p-adic fields. p-Adic Numbers Ultrametric Anal. Appl. 3 (4), 344–358 (2011)
Zúñiga-Galindo, W.A.: Local zeta functions supported on analytic sets and Newton polyhedra. Int. Math. Res. Not. IMRN (15), 2855–2898 (2009)
Zúñiga-Galindo, W.A.: Parabolic equations and Markov processes over p-adic fields. Potential Anal. 28 (2), 185–200 (2008)
Zúñiga-Galindo, W.A.: Fundamental solutions of pseudo-differential operators over p-adic fields. Rend. Sem. Mat. Univ. Padova 109, 241–245 (2003)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Zúñiga-Galindo, W.A. (2016). Pseudodifferential Equations of Klein-Gordon Type. In: Pseudodifferential Equations Over Non-Archimedean Spaces. Lecture Notes in Mathematics, vol 2174. Springer, Cham. https://doi.org/10.1007/978-3-319-46738-2_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-46738-2_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-46737-5
Online ISBN: 978-3-319-46738-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)