P1: JZP 521 86300 Printer: cupusbw 0521863007pre May 17, 2006 This page intentionally left blank 18:4 Char Count= P1: JZP 521 86300 Printer: cupusbw 0521863007pre May 17, 2006 18:4 Char Count= CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 100 MARKOV PROCESSES, GAUSSIAN PROCESSES, AND LOCAL TIMES Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated Gaussian processes It builds to this material through self-contained but harmonized “mini-courses” on the relevant ingredients, which assume only knowledge of measuretheoretic probability The streamlined selection of topics creates an easy entrance for students and experts in related fields The book starts by developing the fundamentals of Markov process theory and then of Gaussian process theory, including sample path properties It then proceeds to more advanced results, bringing the reader to the heart of contemporary research It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in Gaussian process theory This original, readable book will appeal to both researchers and advanced graduate students i P1: JZP 521 86300 Printer: cupusbw 0521863007pre May 17, 2006 18:4 Char Count= Cambridge Studies in Advanced Mathematics Editorial Board: Bela Bollobas, William Fulton, Anatole Katok, Frances Kirwan, Peter Sarnak, Barry Simon, Burt Totaro All the titles listed below can be obtained from good booksellers or from Cambridge University Press For a complete series listing, visit http://www.cambridge.org/us/mathematics Recently published 71 72 73 74 75 76 77 78 79 81 82 83 84 85 86 87 89 90 91 92 93 95 96 97 98 99 R Blei Analysis in Integer and Fractional Dimensions F Borceux & G Janelidze Galois Theories B Bollobas Random Graphs 2nd Edition R M Dudley Real 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Schechter An Introduction to Nonlinear Analysis R Carter Lie Algebras of Finite and Affine Type H L Montgomery & R C Vaughan Multiplicative Number Theory I Chavel Riemannian Geometry D Goldfeld Automorphic Forms and L-Functions for the Group GL(n,R) ii P1: JZP 521 86300 Printer: cupusbw 0521863007pre May 17, 2006 18:4 Char Count= MARKOV PROCESSES, GAUSSIAN PROCESSES, AND LOCAL TIMES M I C H A EL B MA R C U S City College and the CUNY Graduate Center JA Y R O S E N College of Staten Island and the CUNY Graduate Center iii cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521863001 © Michael B Marcus and Jay Rosen 2006 This publication is in copyright Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press First published in print format 2006 isbn-13 isbn-10 978-0-511-24696-8 eBook (NetLibrary) 0-511-24696-X eBook (NetLibrary) isbn-13 isbn-10 978-0-521-86300-1 hardback 0-521-86300-7 hardback Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate To our wives Jane Marcus and Sara Rosen Contents Introduction 1.1 Preliminaries page Brownian motion and Ray–Knight Theorems 2.1 Brownian motion 2.2 The Markov property 2.3 Standard augmentation 2.4 Brownian local time 2.5 Terminal times 2.6 The First Ray–Knight Theorem 2.7 The Second Ray–Knight Theorem 2.8 Ray’s Theorem 2.9 Applications of the Ray–Knight Theorems 2.10 Notes and references Markov processes and local times 3.1 The Markov property 3.2 The strong Markov property 3.3 Strongly symmetric Borel right processes 3.4 Continuous potential densities 3.5 Killing a process at an exponential time 3.6 Local times 3.7 Jointly continuous local times 3.8 Calculating uT0 and uτ (λ) 3.9 The h-transform 3.10 Moment generating functions of local times 3.11 Notes and references 62 62 67 73 78 81 83 98 105 109 115 119 Constructing Markov processes 4.1 Feller processes 4.2 L´vy processes e 121 121 135 vii 11 11 19 28 31 42 48 53 56 58 61 Contents viii 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 Diffusions Left limits and quasi left continuity Killing at a terminal time Continuous local times and potential densities Constructing Ray semigroups and Ray processes Local Borel right processes Supermedian functions Extension Theorem Notes and references 144 147 152 162 164 178 182 184 188 Basic properties of Gaussian processes 5.1 Definitions and some simple properties 5.2 Moment generating functions 5.3 Zero–one laws and the oscillation function 5.4 Concentration inequalities 5.5 Comparison theorems 5.6 Processes with stationary increments 5.7 Notes and references 189 189 198 203 214 227 235 240 Continuity and boundedness of Gaussian processes 6.1 Sufficient conditions in terms of metric entropy 6.2 Necessary conditions in terms of metric entropy 6.3 Conditions in terms of majorizing measures 6.4 Simple criteria for continuity 6.5 Notes and references 243 244 250 255 270 280 Moduli of continuity for Gaussian processes 7.1 General results 7.2 Processes on Rn 7.3 Processes with spectral densities 7.4 Local moduli of associated processes 7.5 Gaussian lacunary series 7.6 Exact moduli of continuity 7.7 Squares of Gaussian processes 7.8 Notes and references 282 282 297 317 324 336 347 356 361 Isomorphism Theorems 8.1 Isomorphism theorems of Eisenbaum and Dynkin 8.2 The Generalized Second Ray–Knight Theorem 8.3 Combinatorial proofs 8.4 Additional proofs 8.5 Notes and references 362 362 370 380 390 394 606 References Fitzsimmons, P and Pitman, J (1999) Kac’s moment formula for aditive functionals of a Markov process Stochastic Process Appl., 79, 117–134 [61, 120] Fukushima, M., Oshima, Y., and Takeda, M (1994) Dirichlet Forms and Symmetric Markov Processes New York: Walter de Gruyter [119] Garsia, A M., Rodemich, E., and Rumsey, Jr., H (1970) A real variable lemma and the continuity of paths of some Gaussian processes Indiana Math J., 20, 565–578 [280, 361] Getoor, R (1975) Markov Processes: Ray Processes and Right Processes New York: Volume 440 of Lecture Notes in Mathematics, Springer-Verlag [119, 188] Getoor, R and Kesten, H (1972) Continuity of local times of Markov processes Comp Math., 24, 277–303 [120] Gin´, E and Kline, R (1975) On quadratic variation of processes with Gause sian increments Ann Probab., 3, 716–721 [495] Griffiths, R C (1984) Characterizations of infinitely divisible multivariate gamma distributions Jour Multivar Anal., 15, 12–20 [6, 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and Shao, Q (2002) A normal comparison inequality and its applications Prob Theory Related Fields, 122, 494–508 [242] Lifshits, M A (1995) Gaussian Random Functions Dordrecht: Kluwer [241] Marcus, M B (1968) Hălder conditions for Gaussian processes with stationo ary increments Trans Amer Math Soc., 134, 2952 [361] Marcus, M B (1970) Hălder conditions for continuous Gaussian processes o Osaka J Math., 7, 483–494 [361] Marcus, M B (1972) Gaussian lacunary series and the modulus of continuity for Gaussian processes Z Wahrscheinlichkeitstheorie und Verw Gebiete, 22, 301–322 [361] Marcus, M B (1991) Rate of growth of local times of strongly symmetric Markov processes In Proceedings Seminar on Stochastic Processes, 1990, volume 24 (pp 253–885) Boston: Birkhauser [455] Marcus, M B (2001) The most visited sites of certain L´vy processes J e Theor Probab., 14, 867–886 [527] Marcus, M B and Pisier, G (1981) Random Fourier Series with Applications to Harmonic Analysis Princeton, NJ: Princeton University Press [241, 280, 281] Marcus, M B and Rosen, J (1992a) Moduli of continuity of local times of strongly symmetric Markov processes via Gaussian processes J Theor Probab., 5, 791–825 [5, 361, 454] Marcus, M B and Rosen, J (1992b) Moment generating functions for the local times of symmetric Markov processes and random walks In Probability in Banach Spaces 8, Proceedings of the Eighth International Conference, Progress in Probability series, volume 30 (pp 364–376) Boston: Birkhauser [579] 608 References Marcus, M B and Rosen, J (1992c) p-variation of the local times of symmetric stable processes and of Gaussian processes with stationary increments Ann Probab., 20, 1685–1713 [495, 496] Marcus, M B and Rosen, J (1992d) Sample path properties of the local times of strongly symmetric Markov processes via Gaussian processes Ann Probab., 20, 1603–1684 [1, 2, 5, 120, 152, 188, 241, 394, 395, 454, 455] Marcus, M B and Rosen, J (1993) φ-variation of the local times of symmetric L´vy processes and stationary Gaussian processes In Seminar e on Stochastic Processes, 1992, volume 33 of Progress in Probability (pp 209–220) Boston: Birkhauser [455, 457, 495] Marcus, M B and Rosen, J (1994b) Laws of the iterated logarithm for the local times of recurrent random walks on Z and of L´vy processes and e recurrent random walks in the domain of attraction of Cauchy random variables Ann Inst H Poincar´ Prob Stat., 30, 467–499 [527] e Marcus, M B and Rosen, J (1994a) Laws of the iterated logarithm for the local times of symmetric L´vy processes and recurrent random walks e Ann Probab., 22, 626–658 [527] Marcus, M B and Rosen, J (1999) Renormalized Self-intersection Local Times and Wick Power Chaos Processes, volume 675 Providence: Memoirs of the AMS [455] Marcus, M B and Rosen, J (2001) Gaussian processes and the local times of symmetric L´vy processes In L´vy Processes–Theory and Applications, e e volume of Math (pp 67–89) Boston: Birkhauser [395] Marcus, M B and Rosen, J (2003) New perspectives on Ray’s theorem for the local times of diffusions Ann Probab., 31, 882–913 [550] Marcus, M B and Shepp, L A (1972) Sample behavior of Gaussian processes In Proceedings of the Sixth Berkeley Symposium Math Statist Prob., volume (pp 423–441) Berkeley: University of California Press [242, 361] McKean, H P (1962) A Hălder condition for Brownian local time J Math o Kyoto Univ., 1, 195–201 [120] McShane, E J and Botts, T A (1959) Real Analysis Princeton, NJ: Van Nostrand [210] Meyer, P.-A (1966) Sur les lois de certaines fonctionnelles multiplicative Publ Inst Statist Univ Paris, 15, 295–310 [120] Millar, P W and Tran, L T (1974) Unbounded local times Z Wahrscheinlichkeitstheorie und Verw Gebiete, 30, 87–92 [120] Molchan, G (1999) Maximum of fractional Brownian motion: Probabilities of small values Comm Math Phys., 205, 97–111 [5, 527] Nisio, M (1967) On the extreme values of Gaussian processes Osaka J Math., 4, 313–326 [361] Perkins, E (1982) Local time is a semimartingale Z Wahrscheinlichkeitstheorie und Verw Gebiete, 60, 79–117 [59, 496] Pitman, E J G (1968) On the behavior of the characteristic function of a probability distribution in the neighbourhood of the origin J Australian Math Soc Series A, 8, 422–443 [594] Port, S C and Stone, C J (1978) Brownian Motion and Classical Potential Theory New York: Academic Press [61] Preston, C (1971) Banach spaces arising from some integral inequalities Indiana Math J., 20, 997–1015 [280] Preston, C (1972) Continuity properties of some Gaussian processes Ann math Statist., 43, 285–292 [280] References 609 Ray, D (1963) Sojourn times of a diffusion processs Ill J Math., 7, 615–630 [5, 49, 58, 59, 61, 120, 144, 188, 537, 550] Reuter, G E H (1969) Remarks on a Markov chain example of Kolmogorov Z Wahrscheinlichkeitstheorie und Verw Gebiete, 13, 315–320 [455] Revuz, D and Yor, M (1991) Continuous Martingales and Brownian Motion New York: Springer-Verlag [56, 61, 123, 125, 173, 581] Rogers, L C G and Williams, D (2000a) Diffusions, Markov Processes, and Martingales Volume Two: Foundations Cambridge: Cambridge University Press [144, 175] Rogers, L C G and Williams, D (2000b) Diffusions, Markov Processes, and Martingales Volume One: Foundations Cambridge: Cambridge University Press [61, 86, 101, 119, 124, 150, 366] Rosen, J (1986) Tanaka’s formula for multiple intersections of planar Brownian motion Stochastic Process Appl., 23, 131–141 [61] Rosen, J (1991) Second order limit laws for the local times of stable processes In S´minaire de Probabilit´s XXV, volume 1485 of Lecture Notes Math e e (pp 407–424) Berlin: Springer-Verlag [496] Rosen, J (1993) p-variation of the local times of stable processes and intersection local time In Seminar on Stochastic Processes, 1991, volume 33 of Progress in Probability (pp 157–168) Boston: Birkhauser [496] Royden, H L (1988) Real Analysis, third edition New York: Macmillan [590] Sato, K (1999) L´vy Processes and Infinitely Divisible Distributions Came bridge: Cambridge University Press [212] Sharpe, M (1988) General Theory of Markov Processes New York: Academic Press [119] Sheppard, P (1985) On the Ray–Knight property of local times J London Math Soc., 31, 377–384 [61, 550] Sirao, T and Watanabe, H (1970) On the upper and lower class for stationary Gaussian processes Trans Amer Math Soc., 147, 301–331 [361] Slepian, D (1962) The one-sided barrier problem for Gaussian noise Bell System Tech J., 41, 463–501 [242] Stroock, D W (1993) Probability Theory: An Analytic View Cambridge: Cambridge University Press [188] Sudakov, V N (1973) A remark on the criterion of continuity of Gaussain sample functions In Proceedings Second Japan-USSR Symposium on Probabability Theory Kyoto, 1972, volume 330 of Lecture Notes in Math., (pp 444–454)., Berlin Springer-Verlag [242] Sudakov, V N and Tsirelson, B S (1978) Extremal properties of half–spaces for spherically invariant measures J Soviet Math., 9, 9–18 Translated from Zap Nauch Sem L.O.M.I 41, 14–24, 1974 [241] Talagrand, M (1987) Continuity of Gaussian processes Acta Math., 159, 99–149 [280] Talagrand, M (1992) A simple proof of the majorizing measure theorem Geom Funct Anal., 2, 118–125 [242, 280] Talagrand, M (2005) The Generic Chaining Berlin: Springer-Verlag [281] Taylor, S J (1972) Exact asymptotic estimates of Brownian path variation Duke Math J., 39, 219–242 [5, 456, 495] Trotter, H (1958) A property of Brownian motion paths Ill J Math., 2, 425–433 [61, 120] van der Hofstad, R., den Hollander, F., and Konig, W (1997) Central limit theorem for the Edwards model Ann Probab., 25, 573–597 [61] 610 References Walsh, J (1978) A diffusion with a discontinuous local time Asterisque, 52, 37–46 [455] Walsh, J (1983) Stochastic integration with respect to local time In Seminar on Stochastic Processes, 1982, volume 16 of Progress in Probability (pp 237–302) Boston: Birkhauser [496, 550] Williams, D (1974) Path decomposition and continuity of local time for one-dimensional diffusion, I Proc London Math Soc, 28, 738–768 [61, 550] Wittman, R (1986) Natural densities for Markov transition probabilities Prob Theory Related Fields, 73, 1–10 [78] Wojtaszczyk, P (1991) Banach Spaces for Analysts Cambridge: Cambridge University Press [591] Index of notation (Ω, F, P ), B, 225 B(t, u), Bd (t, u), 244 BESQ, 368 BESQδ (x), 581 + C(S), Cb (S), Cb (S), Cκ (S), C0 (S), ∞ C0 (S), C0 (S∆ ), 79 Cp , 141 D(u), D(T, d, u), 244 d(s, t), dX (s, t), 8, 194 d(x, A), 24 dX , 243 EY , Eλ , 43 f , f ,y , 32 h, 109 K ⊕ K, ⊕n K, 270 K,8 kL , 110 Lp , Ly , 32 t Lt ,y , 32 M (u), M (T, d, u), 244 m, 189 m(t), 193 N (u), N (T, d, u), 244 N (µ, σ ), 192 P x , 15 P x,0 , 113 611 P x/h , 110 R1 , R+ , R, Rn , S∆ , 63 TA , 24, 70 U α , 16 α UA , 90 α u , 16 uα , 90 A uT , 42 Wf , 209 y ↑↑ x, #S, 233 ∼, ≈, ¯t Ly , 98 ¯ A, 215 ◦θt , 21 cov, 201 det Σ, 191 med, 24 n p, p , G, 23, 326 Gt , 21 γ, γn , 193 L, 93 M, 29, 66 F, Ft , F , 15 F , Ft , 19 h h F , Ft , 112 GT ,GT + , 23 612 Gt , 23 Gt+ , 23 H, 164 M, 29 M+ , M, 164 N , 561 N , IN, σ(u), 270 Φ, φ, 192 Ψ, 404 φ ◦ Φ−1 , 193 ψ(λ), 135 Σ, 189 Σ(s, t), 190, 193 S, 208 σ, 270 σ( · ), Index of notation σ ∗ , σ∗ , 274 σ , σG , 307 τ (s, u),τ (s, A), τA , 43 τz , 39 θt , 21 U, 193 f, Xt , 82 G, Gt , 82 Ω, 82 θt , 82 f , 165 P x , 82 Z, ζ, 72 B(Ω), Bb (Ω), Author index Adler, R J., 241 Ahlfors, L., 142 Donsker, M., 527 Doob, J L., 101, 237 Dudley, R M., 4, 5, 215, 238, 241, 244, 245, 280, 456, 495 Dynkin, E B., 394 Babkov, S., 242 Bakry, D., 242 Bapat, R B., 6, 579 Barlow, M., 2, 120, 455 Bass, R F., 5, 59, 526 Ba˜uelos, R., 527 n Belyaev, Yu K., 213, 241 Berman, N., 579 Bertoin, J., 142, 188, 527 Biane, P., 61, 550 Billingsley, P., 592 Bingham, N., 304, 438, 594 Blackburn, R., 527 Blumenthal, R., 61, 119, 120 Boas, R., 349 Borell, C., 214, 241 Botts, T A., 210 Boylan, E., 120 Brydges, D., Ehrhard, A., 241 Eisenbaum, N., 1, 2, 5, 6, 60, 61, 394, 527, 534, 550, 579 Erdăs, P., 361 o Feller, W., 91, 141, 166, 382, 565, 566, 594, 596 Fernique, X., 4, 238, 241, 242, 244, 251, 280, 361 Fitzsimmons, P., 61, 120, 188 Frăhlich, J., o Fukushima, M , 119 Garcia, A M., 280, 361 Getoor, R., 61, 119, 120, 188 Goldie, C., 304, 438, 594 Griffin, P., 59, 526 Griffiths, R C., 6, 579 Caballero, R., 527 Chung, K L., 61, 119, 361, 455, 592 Hawkes, J., 2, 120, 455 Horn, R A., 579 Hu, Y., 60 de la Vega, F., 495 Dellacherie, C., 73, 119, 180, 188 den Hollander, F., 60 Donoghue, W., 236 Ibragamov, I., 142 Ito, K., 58, 213, 241, 593 613 614 Author index Jain, N., 213, 241, 242, 280, 593 Johnson, C R., 579 Kac, M., 61 Kahane, J.-P., 229 Kallenberg, O., 188 Kallianpur, G., 213, 241 Karamata, J., 594 Kaspi, H., 2, 6, 61, 395, 579 Katzenelson, Y., 336 Kawada, T, 495 Kesten, H., 120 Khoshnevisan, D., 188, 527 Knight, F., 49, 58, 61 Kolmogorov, A N., 243, 441, 455 Konig, W., 60 Kono, N., 361, 495 Lacey, M., 527 Landau, H J., 242 Le Jan, Y., 119 Ledoux, M., 4, 241, 242, 280, 580, 591 Li, W., 242 Lifshits, M A., 241 Linnik, Y., 142 L´vy, P., 61, 456 e Maisonneuve, B., 119 Marcus, M B., 1, 2, 5, 61, 120, 152, 188, 241, 242, 280, 361, 394, 454, 455, 457, 495, 527, 550, 579, 593 McKean, H., 58, 120 McShane, E J., 210 Meyer, P.-A., 73, 119, 120, 180, 188 Millar, P W., 120 Molchan, G., 5, 527 Nisio, M., 213, 241, 361, 593 Oshima, Y., 119 Perkins, E., 59, 496 Pisier, G., 241, 280, 281 Pitman, E J G., 594 Pitman, J., 61, 120 Plemmons, R J., 579 Port, S C., 61 Preston, C., 280 Ray, D., 5, 49, 58, 59, 61, 120, 144, 188, 537 Reuter, G E H., 455 Revuz, D., 56, 61, 123, 125, 173, 581 Rodemich, E., 280, 361 Rogers, L C G., 61, 86, 101, 119, 124, 144, 150, 175, 366 Rosen, J., 1, 2, 5, 61, 120, 152, 188, 361, 394, 454, 455, 457, 495, 496, 527, 550, 579 Royden, H L., 590 Rumsey Jr., H., 280, 361 Sato, K, 212 Shao, Q., 242 Sharpe, M., 119 Shepp, L A., 242, 361 Sheppard, P., 61, 550 Shi, Z., 2, 5, 60, 61, 395, 527 Sirao, T., 361 Slepian, D., 242 Smits, R G., 527 Spencer, T., Stone, C J., 61 Stroock, D., 188 Sudakov, V N., 214, 233, 241 Takeda, M., 119 Talagrand, M., 4, 234, 241, 242, 244, 263, 280, 580, 591 Taylor, S J., 5, 456, 495 Teugels, J., 304, 438, 594 Tran, L., 120 Author index Trotter, H., 61, 120 Tsirelson, B S., 214, 241 van der Hofstad, R , 60 Varadhan, S R S., 527 Walsh, J., 455, 496, 550 Watanabe, H., 361 615 Williams, D., 61, 86, 101, 119, 124, 144, 150, 175, 366, 550 Wittman, R., 78 Wojtaszczyk, P., 591 Yor, M., 56, 61, 123, 125, 173, 550, 581 Subject index Abelian group, 235 analytic set, 583 approximate δ-function, approximate identity, associated Gaussian process, 76, 194, 324, 396, 398, 551 associated process, 76, 194, 398, 551 family of, 396 asymptotic functions, augmented filtration, 67 canonical version, 15 killed, 157 local time of, 31 planar, 534 quadratic variation of, 457 two dimensional, 534 CAF, see continuous additive functional Cameron–Martin Formula, 516 canonical Gaussian measure, 193, 214 canonical stable process, 141 Category (1), 326 cemetery state, 63 Chapman–Kolmogorov equation, 12, 13, 77 characteristic function, 189 class G, 326 class S, 208 comparable functions, concentration inequality, 224 continuous additive functional, 32, 43, 83 potential of, 90 potential operator of, 90 contraction property, 125 contraction resolvent, 125, 126, 128 strongly continuous, 587 Banach–Mazur Theorem, 591 Belyaev dichotomy, 213 Bessel process, 158, 581 squared, 581 Blumenthal Zero–One Law, 26, 70 Boas’s Lemma, 597 Bochner integral, 587 Bochner’s Theorem, 236 Borel right process, 67 local, 179, 551 Borel semigroup, 63 Borell, Sudakov–Tsirelson Theorem, 214 bounded discontinuity, 208 branch point, 170 Brownian motion, 11, 13, 276 most visited sites of, 510 616 Subject index contraction semigroup strongly continuous, 122, 588 counting measure, 184 covariance kernel, 203 cylinder sets, 20, 398 death time, 72 diameter of a metric space, 244 diffusion, 144, 530 distinguishable set, 233 Doob’s Lp inequality, 86, 101 Dudley’s metric entropy condition, 238, 245, 249, 251 dyadic numbers, 580 Dynkin Isomorphism Theorem, 364, 380 Edwards model of polymers, 60 Eisenbaum Isomorphism Theorem, 362, 383, 454 excessive function, 182 excessive regularization, 165 exponential random variable, 202 favorite point, see most visited sites 497 Feller process, 121 Feller semigroup, 121 Fernique’s Lemma, 238 filtered, filtration, first hitting time, 24, 70, 151 Fourier transform, fractional Brownian motion, 276, 497 scaling property, 498 Gaussian lacunary series, 336 Gaussian Markov process, 195 Gaussian metric, 233 Gaussian process, 193 p-variation of, 457 617 associated, 76, 194, 270, 398, 551 Banach space–valued, 225 boundedness, 255, 263 class G, 326 continuity, 255, 267 covariance kernel of, 193 mean function of, 193 metric for, 233 modulus of continuity for, 258, 283 periodic, 280 with infinitely divisible squares, 561 Gaussian random variable, 189 characteristic function of, 189 covariance matrix of, 189 mean vector of, 189 with infinitely divisible squares, 560 Gaussian sequence, 193 Generalized Second Ray–Knight Theorem, 455 generator, 556 group, 235 h-transform, 109, 112 h-transform process, 112 Hille–Yosida Theorem, 587 hitting time first, 24, 70, 151 measurability of, 70 Hunt process, 120, 152 Hunt’s Theorem, 127 increments variance, 307, 317 infinitely divisible process, 135 infinitely divisible squares, 560 inverse local time, 39, 97 of a symmetric stable process, 502 618 Subject index inverse time, 105 irreducible matrix, 600 isomorphism theorems, 49 isoperimetric inequality, 214 Kac’s Moment Formula, 44, 49, 74, 116, 117 Karhunen–Lo´ve expansion, 206 e Khintchine’s law of the iterated logarithm, 15, 431 killed process, 81, 152 killing operator, 110 Kolmogorov Construction Theorem, 14, 122, 173 Kolmogorov’s Theorem for path continuity, 580 lacunary series, 336 Laplace transform, 565 law of the iterated logarithm, 15, 424, 470 left continuous inverse, 270 left-hand limit, 8, 147 local time, 62, 83 continuity of, 410 Brownian, 31 inverse, 39, 93 joint continuity, 98 jointly continuous, 32, 58, 99 modulus of continuity for, 59 moment generating function of, 115 normalization of, 92 of a Markov process, 98 p-variation of, 479 quadratic variation of, 59 stochastic integral representation, 56 total accumulated, 56, 99 local uniform continuity, local uniform convergence, locally homogeneous, 421 locally homogeneous metric space, 359 Lusin’s Theorem, 585 L´vy exponent, 135 e L´vy measure, 330 e L´vy process, 135, 212 e without a Gaussian component, 139 L´vy’s inequality, 591 e L´vy’s uniform modulus of cone tinuity, 15 L´vy–Ito–Nisio Theorem, 591 e L´vy–Khintchine Theorem, 136 e M –matrix, 561, 601 majorant, 274, 297 majorizing measure, 256 Markov kernel, 62 Markov process, 62 Gaussian, 195 right continuous simple, 64 simple, 63 standard, 151 Markov property, 19, 62 local strong, 182 simple, 20, 21, 64 strong, 24, 67 Markov semigroup, 63 measurable space, median, mesh, 456 metric entropy, 244, 271 minorant, 274 modulus of continuity, 15 exact local, 282 exact uniform, 282 for a Gaussian process, 258, 283 m-, 292 moment generating function, 115 Subject index Monotone Density Theorem, 595 most visited sites, 59, 497, 510, 525 negative, negative off-diagonal elements, 552 non-decreasing rearrangement, 271 normal random variable, 192 normalized regularly varying function, 304 occupation density formula, 99 occupation measure, 33 operator norm, 457 optional σ-algebra, 174 optional process, 175 optional set, 175 Ornstein–Uhlenbeck process, 57 oscillation function, 209 packing number, 244 Paley–Zygmund Lemma, 597 Parseval’s Theorem, 10 partition, 456, 458 pathwise connected space, 567 permutation matrix, 600 positive, positive definite, 76, 78 function, 190 matrix, 190 positive matrix, 552, 561 positive maximum principle, 127 positive operator, 122, 127 positive row sums, 552 potential, 183 of a continuous additive functional, 90 potential density, 17, 77, 105, 125, 530 potential operator, 16, 65, 125 of a continuous additive functional, 90 619 probability space, Projection Theorem, 583 pseudo-metric, pure jump process, 139 p-variation, 457 quadratic variation, 59, 457 quasi left continuity, 149 random Fourier series, 238 random variable, random walk, 60 rapidly decreasing, 513 rapidly decreasing function, 545 Ray process, 172 Ray semigroup, 164, 165 Ray’s Theorem Eisenbaum’s version, 534 for Brownian motion, 56, 543 for diffusions, 532 original version, 538 Ray–Knight compactification, 188 Ray–Knight Theorem applications of, 58 first, 48, 366 second, 53, 372, 378, 386 recurrence of elements, 93 of processes, 95 recurrent process, 17 reducible matrix, 600 reference measure, 73 reflection principle, 27, 158 regular symmetric transition densities, 77 regular transition densities, 76 regularly varying function, 304, 594 reproducing kernel Hilbert space, 204 of fractional Brownian motion, 511 620 Subject index resolvent equation, 66, 73, 77 scaling property of fractional Brownian motion, 498 semigroup property, 16 separability, shift operator, 21 signature matrix, 561 Slepian’s Lemma, 227, 295 slowly varying function, 304, 594 smoothly varying function, 438 Souslin scheme, 583 spectral density, 237, 238 spectral distribution, 237, 238, 276 speed measure, 144 squared Bessel process, 581 stable mixture, 427, 437 stable process, 141 canonical, 42, 141 standard L2 metric, 233 standard augmentation, 28, 69 standard normal sequence, 192 state space, 63 stationary increments, process with, 235 stationary process, 235 stochastic continuity, stochastic process, Stone–Weierstrass Theorems, 590 stopping time, 23, 93 measurability of, 151 strictly negative, strictly positive, strictly positive definite matrix, 190, 598 strong continuity, 176 strongly symmetric process, 75 sub-Markov kernel, 62 sub-Markov semigroup, 63 supermedian function, 164, 182 surface measure, 214 symmetric operator, 130 symmetric process, 75 symmetric stable process inverse local time, 502 Talagrand’s Theorem, 263, 267 terminal time, 42, 105 tight family of measures, 591 transience of elements, 93 of processes, 95 transition operator, 16 transition probability density function, 16 transition semigroup, 63, 64 type A, 307 u-distinguishable, 233 unbounded discontinuity, 208 uniform norm, 121 usual augmentation, see standard augmentation vector lattice, 590 version, 98 weakly diagonally dominant, 556 zero row sums, 556 ... Although some inkling of this idea appeared earlier in Brydges, Frăhlich and Spencer o (1982) we think that credit for formulating it in an intriguing and usable format is due to E B Dynkin (1983),... Z denotes the integers both positive and negative and IN or sometimes N denotes the the positive integers including R1 denotes the real line and R+ the positive half line (including zero) R denotes... iii cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge cb2 2ru, UK Published in the