Markov Processes, Gaussian Processes, and Local Times

1 Introduction page 1(10) 1.1 Preliminaries 6(5) 2 Brownian motion and Ray Knight Theorems 11(51) 2.1 Brownian motion 11(8) 2.2 The Markov property 19(9) 2.3 Standard augmentation 28(3) 2.4 Brownian local time 31(11) 2.5 Terminal times 42(6) 2.6 The First Ray Knight Theorem 48(5) 2.7 The Second Ray Knight Theorem 53(3) 2.8 Ray's Theorem 56(2) 2.9 Applications of the Ray Knight Theorems 58(3) 2.10 Notes and references 61(1) 3 Markov processes and local times 62(59) 3.1 The Markov property 62(5) 3.2 The strong Markov property 67(6) 3.3 Strongly symmetric Borel right processes 73(5) 3.4 Continuous potential densities 78(3) 3.5 Killing a process at an exponential time 81(2) 3.6 Local times 83(15) 3.7 Jointly continuous local times 98(7) 3.8 Calculating uTo and uτ(λ) 105(4) 3.9 The h-transform 109(6) 3.10 Moment generating functions of local times 115(4) 3.11 Notes and references 119(2) 4 Constructing Markov processes 121(68) 4.1 Feller processes 121(14) 4.2 L processes 135(9) 4.3 Diffusions 144(3) 4.4 Left limits and quasi left continuity 147(5) 4.5 Killing at a terminal time 152(10) 4.6 Continuous local times and potential densities 162(2) 4.7 Constructing Ray semigroups and Ray processes 164(14) 4.8 Local Borel right processes 178(4) 4.9 Supermedian functions 182(2) 4.10 Extension Theorem 184(4) 4.11 Notes and references 188(1) 5 Basic properties of Gaussian processes 189(54) 5.1 Definitions and some simple properties 189(9) 5.2 Moment generating functions 198(5) 5.3 Zero-one laws and the oscillation function 203(11) 5.4 Concentration inequalities 214(13) 5.5 Comparison theorems 227(8) 5.6 Processes with stationary increments 235(5) 5.7 Notes and references 240(3) 6 Continuity and boundedness of Gaussian processes 243(39) 6.1 Sufficient conditions in terms of metric entropy 244(6) 6.2 Necessary conditions in terms of metric entropy 250(5) 6.3 Conditions in terms of majorizing measures 255(15) 6.4 Simple criteria for continuity 270(10) 6.5 Notes and references 280(2) 7 Moduli of continuity for Gaussian processes 282(80) 7.1 General results 282(15) 7.2 Processes on Rn 297(20) 7.3 Processes with spectral densities 317(7) 7.4 Local moduli of associated processes 324(12) 7.5 Gaussian lacunary series 336(11) 7.6 Exact moduli of continuity 347(9) 7.7 Squares of Gaussian processes 356(5) 7.8 Notes and references 361(1) 8 Isomorphism Theorems 362(34) 8.1 Isomorphism theorems of Eisenbaum and Dynkin 362(8) 8.2 The Generalized Second Ray-Knight Theorem 370(10) 8.3 Combinatorial proofs 380(10) 8.4 Additional proofs 390(4) 8.5 Notes caul references 394(2) 9 Sample path properties of local times 396(60) 9.1 Bounded discontinuities 396(7) 9.2 A necessary condition for unboundedness 403(3) 9.3 Sufficient conditions for continuity 406(4) 9.4 Continuity and boundedness of local times 410(7) 9.5 Moduli of continuity 417(20) 9.6 Stable mixtures 437(4) 9.7 Local times for certain Markov chains 441(6) 9.8 Rate of growth of unbounded local times 447(7) 9.9 Notes and references 454(2) 10 p-variation 456(41) 10.1 Quadratic variation of Brownian motion 456(1) 10.2 p-variation of Gaussian processes 457(10) 10.3 Additional variational results for Gaussian processes 467(12) 10.4 p-variation of local times 479(3) 10.5 Additional variational results for local times 482(13) 10.6 Notes and references 495(2) 11 Most visited sites of symmetric stable processes 497(33) 11.1 Preliminaries 497(7) 11.2 Most visited sites of Brownian motion 504(7) 11.3 Reproducing kernel Hilbert spaces 511(5) 11.4 The Cameron-Martin Formula 516(3) 11.5 Fractional Brownian motion 519(4) 11.6 Most visited sites of symmetric stable processes 523(3) 11.7 Notes and references 526(4) 12 Local times of diffusions 530(21) 12.1 Ray's Theorem for diffusions 530(4) 12.2 Eisenbaum's version of Ray's Theorem 534(3) 12.3 Ray's original theorem 537(6) 12.4 Markov property of local times of diffusions 543(6) 12.5 Local limit laws for h-transforms of diffusions 549(1) 12.6 Notes and references 550(1) 13 Associated Gaussian processes 551(29) 13.1 Associated Gaussian processes 552(8) 13.2 Infinitely divisible squares 560(10) 13.3 Infinitely divisible squares and associated processes 570(8) 13.4 Additional results about M-matrices 578(1) 13.5 Notes and references 579(1) 14 Appendix 580(23) 14.1 Kolmogorov's Theorem for path continuity 580(1) 14.2 Bessel processes 581(2) 14.3 Analytic sets and the Projection Theorem 583(4) 14.4 Hille-Yosida Theorem 587(2) 14.5 Stone Weierstrass Theorems 589(1) 14.6 Independent random variables 590(4) 14.7 Regularly varying functions 594(2) 14.8 Some useful inequalities 596(2) 14.9 Some linear algebra 598(5) References 603(8) Index of notation 611(2) Author index 613(3) Subject index 616