Uniqueness and Non-Uniqueness of Singular Diffusion Operators

by Eberle, Andreas

• ISBN13: 9783540666288
• ISBN10: 3540666281
• Format: Paperback
• Copyright: 1999-11-01
• Publisher: Springer Verlag

Summary

This volume constitutes the thoroughly refereed post-workshop proceedings of an international workshop on fuzzy logic in Artificial Intelligence held in Negoya, Japan during IJCAI '97.The 17 revised full papers presented have gone through two rounds of reviewing and revision. Three papers by leading authorities in the area are devoted to the general relevance of fuzzy logic and fuzzy sets to AI. The remaining papers address various relevant issues ranging from theory to application in areas like knowledge representation, induction, logic programming, robotics, pattern recognition, etc.

Table of Contents

Introduction	1	(1)
General introduction	1	(4)
Results	5	(4)
Uniqueness problems in various contexts	9	(32)
Uniqueness of the martingale problem	10	(7)
Cores	17	(4)
Essential self-adjointness	21	(1)
Uniqueness of Dirichelt forms	22	(2)
Relations between the notions of uniqueness	24	(6)
Appendix
Existence and uniqueness of C0 semigroups on Banach spaces	30	(5)
Diffusion operators on Lp spaces	35	(6)
Lp uniqueness in finite dimensions	41	(48)
The regular one-dimensional case	42	(9)
Examples and counterexamples I: Regular operators	51	(4)
Diffusion operators on Lp (Rn; dx)	51	(2)
Operators on Lp (R1; e-x2/2 dx)	53	(1)
Perturbations of generalized Schrodinger operators on R1	54	(1)
Regular diffusion operators on Rn	55	(4)
Examples and counterexamples II: Singular operators	59	(7)
Singular generalized Schrodinger operators on R1	59	(2)
Rotationally invariant generalized Schrodinger operators	61	(4)
Degeneracy of second order coefficients	65	(1)
The singular one-dimensional case	66	(9)
Singular diffusion operators on Rn	75	(10)
Appendix
Regularity of distributional solutions of O.D.E.	85	(4)
Markov uniqueness	89	(80)
Weak Sobolev spaces on Rn and on Banach spaces	92	(12)
Weak and strong Sobolev spaces on general state spaces	104	(8)
Maximal Dirichlet extensions	112	(6)
Markov uniqueness in the one-dimensional case	118	(6)
Density of smooth functions in weak Sobolev spaces over Rn	124	(5)
Markov uniqueness in the finite dimensional case	129	(7)
Ergodicity and extremality of symmetrizing measures	136	(11)
Appendix
The geometry of diffusion operators	147	(22)
Generalized differentials	148	(1)
Measurable co-tangent bundles and differentials corresponding to diffusion operators	149	(3)
Diffusion operators on manifolds and vector spaces	152	(1)
Ornstein-Uhlenbeck geometries on path and loop spaces	153	(2)
Horizontal and vertical measure-valued diffusions	155	(4)
Divergence operators and symmetrizing measures	159	(3)
A representation theorem for diffusion operators on L2 spaces	162	(7)
Probabilistic aspects of Lp and Markov uniqueness	169	(16)
Feller classification and uniqueness	170	(8)
Conservativity, ergodicity, and Markov uniqueness	178	(1)
Probabilistic explanations for Lp uniqueness	179	(6)
First steps in infinite dimensions	185	(66)
Infinite dimensional diffusion operators on linear spaces	187	(10)
The generator of the Brownian string --- A counterexample	197	(18)
Markov uniqueness of projective limits	215	(10)
Approximative approaches to uniqueness and existence in Lp	225	(9)
Applications to lattice systems in statistical mechanics	234	(4)
Stability of Lp uniqueness under H--valued perturbations	238	(9)
The general perturbation result	239	(1)
Perturbations of operators with linear drift	240	(5)
Infinite dimensional generalized Schrodinger operators	245	(2)
Applications to perturbed operators	247	(4)
Finite volume quantum fields	247	(3)
Perturbations of the Ornstein-Uhlenbeck operator	250	(1)
The Brownian string in a velocity field	251