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Stochastic Differential Equations of Pure-Jumps

Masao Nagasawa

Published Jan 1, 2000• 0 citations

Abstract

In the preceding chapter we have formulated the equation of motion of relativistic quantum particles as a pair of evolution equations with fractional power generators which are in formal duality, and shown that the evolution equation has an equivalent formulation in terms of stochastic differential equations of pure-jumps. In this chapter the existence and uniqueness of solutions of the stochastic differential equations of pure-jumps will be proved, and then pure-jump Markov processes will be constructed. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

DOI: 10.1007/978-3-0348-8383-2_8

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Stochastic processes in quantum physics

正雄長沢

I Markov Processes.- 1.1 Classical Mechanics.- 1.2 Movement of a Particle with Noise.- 1.3 Transition Probability and the Markov Property.- 1.4 Diffusion Equations.- 1.5 Brownian Motions.- 1.6 The Ito formula.- Appendix. Monotone Lemmas.- II Time Reversal and Duality.- 2.1 Time Reversal of Markov Processes and Duality.- 2.2 Space-Time Markov Processes and Space-Time Duality.- 2.3 Time Reversal and Schroedinger's Representation.- III Non-Relativistic Quantum Theory.- 3.1 Non-Relativistic Equation of Motion.- 3.2 Stationary States and Eigenvalue Problem.- 3.3 Time Reversal of Diffusion Processes.- 3.4 Duality Relation of Diffusion Processes.- 3.5 Equation of Motion in General Cases.- 3.6 Principle of Superposition of Markov Processes.- 3.7 Non-Relativistic Schroedinger Equation.- 3.8 State Preparations and Measurements.- 3.9 Diffusion or Schroedinger Equations ?.- 3.10 The First Technical Convention.- IV Stationary Schroedinger Processes.- 4.1 Stationary States.- 4.2 One-Dimensional Harmonic Oscillator.- 4.3 An Example in Two-Dimension.- 4.4 Superposition of Eigenfunctions.- 4.5 Further Excited States.- 4.6 Hydrogen Atom.- V Construction of the Schroedinger Processes.- 5.1 The Feynman-Kac Formula.- 5.2 Solving the Equation of Motion.- 5.3 Transformation by Multiplicative Functionals.- 5.4 Renormalization.- 5.5 A Variational Method.- 5.6 The Maruyama-Girsanov Formula.- 5.7 A Lagrangian Formulation.- 5.8 The Second Technical Convention.- VI Markov Processes with Jumps.- 6.1 Poisson and Compound Poisson Processes.- 6.2 Poisson Random measures and Point Processes.- 6.3 Stochastic Integrals with Poisson Point Processes.- 6.4 Levy Processes.- 6.5 Stable Processes.- 6.6 Bochner's Subordination.- 6.7 Duality of Subordinate Semi-Groups.- 6.8 Harmonic Transformation of Subordinate Semi-Groups.- 6.9 Duality of Fractional Powers of Time-Dependent Operators.- VII Relativistic Quantum Particles.- 7.1 A Relativistic Schroedinger Equation for a Spinless Paticle.- 7.2 Equation of Motion for Relativistic Quantum Particles.- 7.3 Stationary States of the Relativistic Schroedinger Equation.- 7.4 Stochastic Processes for Relativistic Spinless Particles.- 7.5 Non-Relativistic Limit.- 7.6 A Diffusion Approximation.- VIII Stochastic Differential Equations of Pure-Jumps.- 8.1 Markov Processes with the Generators of Fractional Power.- 8.2 Stochastic Differential Equations of Pure-Jumps.- 8.3 The Case with no Potential Term.- 8.4 To Solve the Stochastic Differential Equations of Pure-Jumps.- 8.5 To Construct Pure-Jump Markov Processes.- 8.6 A Remark on the Integrability Condition.- IX Variational Principle for Relativistic Quantum Particles.- 9.1 Absolute Continuity.- 9.2 Pure-Jump Markov Processes.- 9.3 A Multiplicative Functional.- 9.4 Renormalization and Variational Principle.- X Time Dependent Subordination and Markov Processes with Jumps.- 10.1 Time-Inhomogeneous Subordination.- 10.2 Lemmas.- 10.3 Stochastic Differential Equation with Jumps.- 10.4 A Formula of Feynman-Kac Type.- 10.5 Markov Processes with Jumps.- Appendix. Integration by Parts Formulae.- XI Concave Majorants of Levy Processes and the Light Cone.- 14.1 The Vertex Process of a Levy Process.- 14.2 Propositions on Random walks.- 14.3 Proof of Propositions on Random Walks.- 14.4 Proof of the main Theorems.- 14.5 Examples.- 14.6 The light Cone.- XII The Locality in Quantum Physics.- 12.1 Historical Overview.- 12.2 Hidden-Variable Theories.- 12.3 Locality of Hidden-Variable Theories.- 12.4 Spin-Correlation of Three Particles.- 12.5 Gudder's Hidden-Variable Theory.- 12.6 Spin-Correlations in Gudder's Theory.- 12.7 Some Remarks.- XIII Micro Statistical Theory.- 13.1 The Source of the Noise.- 13.2 Large Deviations of the Renormalized Processes.- 13.3 The Propagation of Chaos.- 13.4 Micro Statistical Mechanics.- 13.5 Propagation of Chaos of Pure-Jump Processes.- 13.6 Superposition of Movements.- 12.7 A Remark on the Gibbs Distribution.- XIV Processes on Open Time Intervals.- 14.1 Diffusion Processes on Open Time Intervals.- 14.2 Time-Reversed Schroedinger Processes.- 14.3 A Theorem of Jeulin-Yor.- 14.4 Reflecting Brownian Motion.- 14.5 Two-Sided Skorokhod Type Problem.- 14.6 Skorokhod Problem with Singular Drift.- 14.7 The Minimum and Maximum Solutions.- 14.8 The Uniqueness and Non-Uniqueness of Solutions.- 14.9 An Application: The Origin of Universes.- XV Creation and Killing of Particles.- 15.1 Non-Linear Differential Equations.- 15.2 Branching Markov Processes.- 15.3 The Expected Number of Particles.- 15.4 Killing.- XVI The lto Calculus.- 16.1 The Ito Integral.- 16.2 Martingales.- 16.3 The Ito Integral with Local Martingales.- 16.4 Ito's formula.- 16.5 Stochastic Differential Equations.- 16.6 Stochastic Differential Calculus.- References.

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