Schrödinger diffusion processes (Probability and its Applications)

✍ Scribed by Robert Aebi

• Publisher: Birkhäuser
• Year: 1996
• Tongue: English
• Leaves: 199
• Series: Probability and its Applications
• Edition: 1
• Category: Library

✦ Synopsis

In 1931 Erwin Schr?dinger considered the following problem: A huge cloud of independent and identical particles with known dynamics is supposed to be observed at finite initial and final times. What is the "most probable" state of the cloud at intermediate times? The present book provides a general yet comprehensive discourse on Schr?dinger's question. Key roles in this investigation are played by conditional diffusion processes, pairs of non-linear integral equations and interacting particles systems. The introductory first chapter gives some historical background, presents the main ideas in a rather simple discrete setting and reveals the meaning of intermediate prediction to quantum mechanics. In order to answer Schr?dinger's question, the book takes three distinct approaches, dealt with in separate chapters: transformation by means of a multiplicative functional, projection by means of relative entropy, and variation of a functional associated to pairs of non-linear integral equations. The book presumes a graduate level of knowledge in mathematics or physics and represents a relevant and demanding application of today's advanced probability theory.

✦ Table of Contents

Cover......Page 1
Schrödinger Diffusion Processes......Page 4
ISBN 3764353864 ISBN 0817653864......Page 5
Contents......Page 6
Preface......Page 8
1 Schrödinger's View of Natural Laws......Page 10
1.1 Most probable realizations......Page 11
1.2 A large deviation approach......Page 15
1.3 Prediction from past and future......Page 17
1.4 An analogy to wave functions......Page 21
1.5 Two representations of diffusions......Page 24
1.6 Identification of drift......Page 27
2 Diffusions with Singular Drift......Page 32
2.1 Schrödinger equations......Page 33
2.2 Non-smooth Schrödinger multipliers......Page 38
2.3 Singular transformation of diffusions......Page 42
2.4 Schrödinger processes......Page 55
3.1 Generators and transition densities......Page 64
3.2 Feynman-Kac integral equations......Page 66
3.3 'Killed' integral equations......Page 74
3.4 Equivalence of solutions......Page 79
4.1 Meaning and generalization......Page 86
4.2 Driving Brownian motion......Page 88
4.3 Driving flows of diffeomorphisms......Page 91
5 Large Deviations......Page 100
5.1 Approximate Sanov property......Page 101
5.2 Csiszar's projection and To-topology......Page 105
6 Interacting Diffusion Processes......Page 124
6.1 Eddington-Schrödinger prediction......Page 125
6.2 Limiting distributions......Page 126
6.3 Propagation of chaos in entropy......Page 132
6.4 Renormalization procedures......Page 151
6.5 Conditions on creation and killing......Page 155
7 Schrödinger Systems......Page 160
7.1 Non-linear integral equations......Page 161
7.2 Product measure endomorphisms......Page 164
7.3 A variational principle for local adjoints......Page 170
7.4 Construction of solutions......Page 180
References......Page 184
Index......Page 194