PARABOLIC EQUATIONS OVER THE FIELD OF $ p$-ADIC NUMBERS

Кочубей, А Н

doi:10.1070/im1992v039n03abeh002247

Cited by 62 publications

(38 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In [21] Varadarajan studied the heat kernel on n−dimensional vector spaces over local fields and division rings. Our results constitute a first step toward the generalization of the results of [18] and [21] to higher dimension.…”

Section: Introductionmentioning

confidence: 97%

“…In dimension one the p−adic heat kernel has been studied extensively [6], [11], [12], [15], [18], [19], [22], A systematic study of the parabolic equations and the related Markov processes was given by Kochubei in [18]- [19]. In [21] Varadarajan studied the heat kernel on n−dimensional vector spaces over local fields and division rings.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, motivated by [3] and [18], we consider the Cauchy problem    ∂u(x,t) ∂t + (f (D, β) u) (x, t) = 0, x ∈ Q n p , n ≥ 1, t ∈ (0, T ]…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parabolic Equations and Markov Processes Over p-Adic Fields

Zúñiga–Galindo

2007

Potential Anal

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Parabolic Equations and Markov Processes Over p-Adic Fields

Zúñiga–Galindo

2007

Potential Anal

View full text Add to dashboard Cite

show abstract

“…was suggested in [10], and its mathematical properties and properties of the associated stochastic processes were studied (see also [108,109,110,98] …”

Section: P-adic Stochastic Processesmentioning

confidence: 99%

On p-adic mathematical physics

Драгович¹,

Khrennikov

Козырев

et al. 2009

P-Adic Num Ultrametr Anal Appl

279

167

View full text Add to dashboard Cite

A brief review of some selected topics in p-adic mathematical physics is presented.1 Numbers: Rational, Real, p-AdicWe present a brief review of some selected topics in p-adic mathematical physics. More details can be found in the references below and the other references are mainly contained therein. We hope that this brief introduction to some aspects of p-adic mathematical physics could be helpful for the readers of the first issue of the journal p-Adic Numbers, Ultrametric Analysis and Applications.The notion of numbers is basic not only in mathematics but also in physics and entire science. Most of modern science is based on mathematical analysis over real and complex numbers. However, it is turned out that for exploring complex hierarchical systems it is sometimes more fruitful to use analysis over p-adic numbers and ultrametric spaces. p-Adic numbers (see, e.g. [1]), introduced by Hensel, are widely used in mathematics: in number theory, algebraic geometry, representation theory, algebraic and arithmetical dynamics, and cryptography.The following view how to do science with numbers has been put forward by Volovich in [2,3]. Suppose we have a physical or any other system and we 1 make measurements. To describe results of the measurements, we can always use rationals. To do mathematical analysis, one needs a completion of the field Q of the rational numbers. According to the Ostrowski theorem there are only two kinds of completions of the rationals. They give real R or p-adic Q p number fields, where p is any prime number with corresponding p-adic norm |x| p , which is non-Archimedean. There is an adelic formula p |x| p = 1 valid for rational x which expresses the real norm |x| ∞ in terms of p-adic ones. Any p-adic number x can be represented as a series x = ∞ i=k a i p i , where k is an integer and a i ∈ {0, 1, 2, ..., p − 1} are digits. To build a mathematical model of the system we use real or p-adic numbers or both, depending on the properties of the system [2,3].Superanalysis over real and p-adic numbers has been considered by Vladimirov and Volovich [4,5]. An adelic approach was emphasized by Manin [6].One can argue that at the very small (Planck) scale the geometry of the spacetime should be non-Archimedean [2,3,7]. There should be quantum fluctuations not only of metrics and geometry but even of the number field. Therefore, it was suggested [2] the following number field invariance principle: Fundamental physical laws should be invariant under the change of the number field.One could start from the ring of integers or the Grothendieck schemes. Then rational, real or p-adic numbers should appear through a mechanism of number field symmetry breaking, similar to the Higgs mechanism [8, 9].Recently (for a review, see [10,11,12,13,14]) there have been exciting achievements exploring p-adic, adelic and ultrametric structures in various models of physics: from the spacetime geometry at small scale and strings, via spin glasses and other complex systems, to the universe as a whole. There has been al...

show abstract

“…Various constructions of Markov processes on non-compact perfect locally compact Abelian groups have been developed by Evans [22], Haran [30], [31], Ismagilov [34], Kochubei [39], [40], Albeverio and Karwowski [1], [2], Albeverio and Zhao [3], Del Muto and Figá-Talamanca [43], [44], Rodrigues-Vega and Zuniga-Galindo [67], [51]. They studied X-valued infinitely divisible random variables and processes by using tools of Fourier analysis; for general references, see Hewitt and Ross [32], Taibleson [56] and Kochubei [40].…”

Section: Introductionmentioning

confidence: 99%

Изотропные Марковские Полугруппы На Ультраметрических Пространствах

et al. 2014

View full text Add to dashboard Cite

Let (X, d) be a separable ultra-metric space with compact balls. Given a reference measure µ on X and a distance distribution function σ on [0 , ∞), we construct a symmetric Markov semigroup {P t } t≥0 acting in L 2 (X, µ). Let {X t } be the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator L and its spectrum which is pure point. In the particular case when X = Q n p , where Q p is the field of p-adic numbers, our construction recovers the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigami's jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

show abstract

PARABOLIC EQUATIONS OVER THE FIELD OF $ p$-ADIC NUMBERS

Cited by 62 publications

References 4 publications

Parabolic Equations and Markov Processes Over p-Adic Fields

Parabolic Equations and Markov Processes Over p-Adic Fields

On p-adic mathematical physics

Изотропные Марковские Полугруппы На Ультраметрических Пространствах

Contact Info

Product

Resources

About