In this paper we construct and study the fundamental solution of Cauchy problem for a p-adic parabolic equation where we use pseudodifferential operator on local field defined by Su Weiyi. We also show that the fundamental solution of the equation is the transition density of a Markov process.
REFERENCES
1.
Albeverio
S.
, and Karwoski
W.
, Diffusion in p-adic numbers. In: K.
Ito
, H.
Hida
(Eds.), Gaussian Random Fields
, pp. 86
–99
, World Scientific
, Singapore
, 1991
.2.
Albeverio
S.
, and Karwoski
W.
, A random walk on p-adics: the generator and its spectrum
, Stochastic Process. Appl.
53
(1994
), 1
–22
.3.
Albeverio
S.
, and Kurasov
P.
, Singular perturbations of differential operators, In: Solvable Schrödinger type operators, London Math. Soc. Lecture Note Ser
. 271
, Cambridge Univ. Press
, Cambridge
, 2000
.4.
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. Appl.
12
(2006
), no. 4
, 393
–425
.5.
Albeverio
S.
, Kuzhel
S.
, and Torba
S.
, Harmonic analysis in the p-adic Lizorkin spaces: fractional operators, pseudo-differential equations, p-adic wavelets, Tauberian theorems
. J. Fourier Anal. Appl.
12
(2006
), no. 4
, 393
–425
.6.
Avetisov
A. V.
, Bikulov
A. H.
, Kozyrev
S. V.
, and Osipov
V. A
, p-adic models of ultrametric diffusion constrained by hierarchical energy landscapes
, J. Phys. A: Math. Gen.
35
(2002
), 177
–189
.7.
Avetisov
A. V.
,Bikulov
A. H.
, and Osipov
V. A.
, p-adic description of characteristic relaxation in complex systems
, J. Phys. A: Math. Gen.
36
(2003
), 4239
–4246
.8.
Khrennikov
A.
, p-Adic Valued Distributions in Mathematical Physics, Kluwer
, Dordrechht
, 1994
.9.
Khrennikov
A.
, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer
, Dordrechht
, 1997
.10.
Kochubei
A. N.
, Parabolic equations over the field of p-adic numbers
, Math. USSR Izvestiya
39
(1992
), 1263
–1280
.11.
Kochubei
A. N.
, Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields, Marcel Dekker, Inc
, New York, NY
, 2001
.12.
Qiu
H.
, Su
W. Y.
, Pseudo-differential operators over p-adic fields
. Science in China, Ser.A
, 41
(4
) (2011
),323
–336
.13.
Su
W.Y.
, Psuedo-differential operators and derivatives on locally compact Vilenkin groups
. Science in China, Ser.A
, 35
(7
) (1992
),826
–836
.14.
Taibleson
M.H.
, Fourier Analysis on Local Fields, Princeton University Press
, Princeton
, 1975
.15.
Varadarajan
V. S.
, Path integrals for a class of p-adic Schrödinger equations
, Lett. Math. Phys.
39
(1997
), 97
–106
.16.
Vladimirov
V. S.
, Volovich
I. V.
, and Zelenov
E. I.
, p-adic Analysis and Mathematical Physics, World Scientific
, Singapore
, 1994
.
This content is only available via PDF.
© 2016 Author(s).
2016
Author(s)
You do not currently have access to this content.
