Pseudo-Differential Equations and Stochastics

Monograph and reference providing comprehensive coverage of the recent developments in the theory of non-Archimedian pseudo-differential equations and its applications. Explores elliptic and hyperbolic equations, Green functions and their asymptotics, and spectral theory.

Anatoly N. Kochubei is a Principal Researcher at the Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev

Preface

iii

Preliminaries

1

(34)

Field extensions

1

(2)

Finite fields

3

(2)

Local fields

5

(6)

Polynomials and quadratic forms

11

(3)

Integration and harmonic analysis

14

(3)

Distributions

17

(4)

Homogeneous distributions

21

(3)

Gaussian integrals

24

(7)

Adeles

31

(4)

Fundamental Solutions

35

(56)

The Riesz kernel

35

(3)

The fractional differentiation operator

38

(3)

The family {Dα,α> 0} as a semigroup of operators

41

(5)

The fractional differential operator on ideles and zeroes of Riemann's zeta function

46

(3)

Pseudo-differential operators over a local field

49

(8)

Coordinate representation

57

(4)

Fundamental solutions of elliptic operators

61

(11)

Green functions of elliptic operators

72

(3)

Hyperbolic equations

75

(3)

Cauchy problem for an equation of the Schrodinger type

78

(10)

On some models of p-adic quantum theory

88

(2)

Comments

90

(1)

Spectral Theory

91

(42)

Fractional differentiation operator

91

(3)

An operator of the Schrodinger type

94

(12)

Spectral properties of Dα on open subsets of a local field

106

(14)

An analogue of the Hamiltonian of point interactions

120

(8)

Multiplicative fractional differentiation

128

(2)

Comments

130

(3)

Parabolic Equations

133

(60)

An analogue of the heat equation

133

(10)

Probabilistic interpretation

143

(3)

Stabilization

146

(4)

General uniqueness theorem

150

(2)

Fundamental solutions of parabolic equations

152

(10)

Heat equation on a ball

162

(5)

Heat equation on the group of units

167

(3)

On parabolic pseudo-differential equations over Rn

170

(21)

Comments

191

(2)

Construction of Processes

193

(52)

Notations and preliminaries

193

(2)

Stochastic integrals of deterministic functions

195

(7)

Stochastic integrals of random functions

202

(3)

Stochastic differential equations

205

(7)

Rotation-invariant processes with independent increments

212

(6)

The generator and its spectrum

218

(5)

Recurrence and hitting probabilities

223

(5)

Processes on adeles

228

(3)

General stable distributions

231

(11)

Comments

242

(3)

Analysis over Infinite Extensions of a Local Field

245

(42)

Measurable vector spaces

245

(13)

An infinite extension as a topological vector space

258

(2)

Measure and integration

260

(7)

Fourier transform and fractional differentiation

267

(7)

Spectrum

274

(2)

Heat equation

276

(6)

Hypersingular integral representation

282

(4)

Comments

286

(1)

Processes with Non-Archimedean Time Parameters

287

(16)

Local field Brownian motion

287

(6)

The Wiener process with the p-adic time parameter

293

(9)

Comments

302

(1)

Bibliography

303

(12)

Index

315

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Pseudo-Differential Equations and Stochastics over Non-Archimedean Fields

Summary

Author Biography

Table of Contents

Supplemental Materials