Author	Aebi, Robert. author
Title	Schrรถdinger Diffusion Processes [electronic resource] / by Robert Aebi
Imprint	Basel : Birkhรคuser Basel, 1996
Connect to	http://dx.doi.org/10.1007/978-3-0348-9027-4
Descript	186 p. online resource

In 1931 Erwin Schrรถdinger 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 Schrรถdinger'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 Schrรถdinger'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

1 Schrรถdingerโ{128}{153}s View of Natural Laws -- 1.1 Most probable realizations -- 1.2 A large deviation approach -- 1.3 Prediction from past and future -- 1.4 An analogy to wave functions -- 1.5 Two representations of diffusions -- 1.6 Identification of drift -- 2 Diffusions with Singular Drift -- 2.1 Schrรถdinger equations -- 2.2 Non-smooth Schrรถdinger multipliers -- 2.3 Singular transformation of diffusions -- 2.4 Schrรถdinger processes -- 3 Integral and Diffusion Equations -- 3.1 Generators and transition densities -- 3.2 Feynman-Kac integral equations -- 3.3 โ{128}{152}Killedโ{128}{153} integral equations -- 3.4 Equivalence of solutions -- 4 Itรดโ{128}{153}s Formula for Non-Smooth Functions -- 4.1 Meaning and generalization -- 4.2 Driving Brownian motion -- 4.3 Driving flows of diffeomorphisms -- 5 Large Deviations -- 5.1 Approximate Sanov property -- 5.2 Csiszarโ{128}{153}s projection and ?0-topology -- 6 Interacting Diffusion Processes -- 6.1 Eddington-Schrรถdinger prediction -- 6.2 Limiting distributions -- 6.3 Propagation of chaos in entropy -- 6.4 Renormalization procedures -- 6.5 Conditions on creation and killing -- 7 Schrรถdinger Systems -- 7.1 Non-linear integral equations -- 7.2 Product measure endomorphisms -- 7.3 A variational principle for local adjoints -- 7.4 Construction of solutions -- References