Brownian Motion Peter Morters and Yuval Peres Index More Information

Cambridge University Press 978-0-521-76018-8 - Brownian Motion Peter Morters and Yuval Peres Index More information

Index

adapted, 37 zero set, 48 additive functional, 95 Brownian scaling, 13 Adelman, Omer, 94, 150, 322, 326 Burdzy, Krzysztof, 94, 150, 326, 345, 384 almost surely, 8 Burkholder–Davis–Gundy inequality, 64 arcsine law generalised, 223 Cameron–Martin theorem, 25 for the last sign-change, 136 Cantor set for the last zero, 136 Hausdorff dimension, 115 for the position of the maximum, 136 Minkowski dimension, 115 for the time spent above zero, 137, 215 capacity Azéma–Yor embedding theorem, 129 α-capacity, 113, 234 general definition, 234 Bachelier, Louis, 33 Newtonian, 113 ballot problem, 63 of an edge, 358 Bessel process, 64 Riesz, 113 two-dimensional, 171 Cauchy process, 52 Blumenthal’s 0-1 law, 38 compactness argument, 359 bounded variation, 22 conditional expectation, 48 Brown, Robert, 33 conformal invariance, 5, 202 Brownian bridge, 30, 61, 285 continuity Brownian excursion, 244 Lévy’s modulus, 16 Brownian frontier, see outer boundary local Hölder, 17 Brownian hull, 329 convergence in distribution, 346 Brownian motion convolution, 46 area, 46 cooling, 214 conformal invariance, 202 countable stability property, 97 d-dimensional, 36 countable subadditivity, 100 definition, 7 covering, 96 fractional, 34 cut line, 150 graph, 102 cut point, 150, 328, 344 image, 102 cutset, 359 inifinite multiplicity, 385 Lévy’s construction, 9 Davis, Burgess, 34, 64, 222, 307, 326 linear, 7 derivative monotonicity, 18 upper and lower, 19 non-colliding, 152 differentiability path, 102 in a point, 20 planar, 36 nowhere, 21 range, 102 diffusion reflected, 146 matrix, 2 scaling invariance, 12 dimension, 96 skew-product representation, 206 Dirichlet problem standard,2,7,36 existence, 70 time inversion, 13 harmonic measure, 84 transient, 78, 228 representation of solution, 3 two-sided, 30 solution, 69 with drift, 7 uniqueness, 225

400

Index 401

Dirichlet space, 24, 220 σ-algebra, 38 dissipation rate, 213 Girsanov’s theorem, 64 domain, 65 global maximum, 39 Donsker’s invariance principle, 4, 132 Green kernel, 80 Doob’s Lp maximal inequality, 355 Green’s function Doob’s weak maximal inequality, 355 definition, 80 Doob, Joe, 64 properties, 81 downcrossing, 153 drift, 7 harmonic function vector, 2 definition, 65 Dubins’ embedding theorem, 127 history, 253 Dvoretzky, Aryeh, 34, 73, 94, 150, 287, 307, 326 maximum principle, 67 Dvoretzky–Erdos˝ test, 73 mean value property, 65 Dynkin, Eugene, 63 removable, 253 harmonic measure Einstein, Albert, 33 conformal invariance, 204 embedding definition, 84 Azéma–Yor, 129 from infinity, 87 Dubins, 127 on the sphere, 85 Skorokhod, 127 support, 87 energy Harnack principle, 84 α-energy, 108, 234 Harris inequality, 123 energy method Hausdorff content, 99 terminology, 108 Hausdorff dimension, 4 envelope and measure, 100 lower, 73 countable stability, 115 upper, 118 definition, 99 equilibrium measure, 228, 229 of graph, 110 equivalent measure, 25 of range, 110 Erdos,˝ Paul, 34, 73, 94, 150, 221, 287 of zero set, 108 Erdos–Kac˝ theorem, 221 Hausdorff gauge, see gauge function Evans, Steven N., 150, 288, 313, 326 Hausdorff measure exceptional set definition, 99 of times, 20 generalised, 178 exchangeable, 19 Hausdorff, Felix, 116 excursion, 148, 244 Hawkes’ theorem, 259 exit from an interval, 217 heat equation, 213 exit time, 57, 62 heating, 214 Helly-Bray theorem, 348 fast time, 291 Hewitt–Savage 0-1 law, 19 Feynman–Kac formula, 214, 216 hitting probability, 72, 236 filtered probability space, 37 Hölder continuous, 101 filtration, 37 hull, 329, 336 complete, 190 convention, 42, 190 increasing event, 147 finite-dimensional distributions, 8 independent, 37 first arcsine law, 136 independent increments, 1, 7 first moment method, 259 initial distribution, 2 FKG inequality, 151 integral test, 73 flow, 358 intersection equivalence fractional Brownian motion, 34 definition, 263 Frostman’s lemma, 111, 117 intersection exponents, 288, 327, 343, 344, 345 Frostman, Otto, 117, 253 intersections functional central limit theorem, 132 dimension, 261 existence, 255, 257, 261 Galton–Watson tree, 360 nonexistence, 255, 257 gauge function, 117, 178, 287 intrinsic, 96 Gaussian inversion at the circle, 86 random vector, 9 irregular point, 224 standard, 349 Itô’s formula process, 34 multidimensional, 200 geometric distribution, 154 nonsmooth case, 210 germ one-dimensional, 196 0-1 law, 38 with additional dependence, 197

402 Index

Itô, Kiyoshi, 187, 195, 209, 258 uniformly integrable, 353 martingale inequality, 355 Kahane, Jean-Pierre, 33, 287, 326 mass distribution, 105 Kakutani’s theorem, 234 mass distribution principle, 105 Kakutani, Shizuo, 6, 30, 34, 94, 150, 234, 253, 287 max-flow min-cut theorem, 359 Kallianpur–Robbins law, 93, 95 McKean’s theorem, 102, 114 Kaufman’s theorem, 115, 279 McKean, Henry, 102, 114, 117, 222, 258, 287 Kelvin transform, 94 microscopic, 1 kernel, 232 Minkowski dimension Khoshnevisan, Davar, 287, 302, 325, 384 lower, 97 Kochen–Stone lemma, 74, 91 upper, 97 Kolmogorov, A.N., 33, 133, 149, 223, 240 modification, 194 modulus normal distribution, 49 Laplace transform, 348 modulus of continuity, 14, 16, 31 Laplace’s method, 207 monotonicity, 100 law of in an interval, 18 maximum, 45 multiple points law of large numbers, 14 existence, 272 law of the iterated logarithm nonexistence, 272 failure, 290 for Brownian motion, 118 neighbourhood recurrent, 72 for general random walk, 127 nonpolar set, 85 for simple random walk, 121 nowhere differentiable, 21 Hartman and Wintner, 127, 149 Khinchin, 119, 121, 149 optional stopping theorem, 54 Kolmogorov, 149 Orey and Taylor’s theorem, 291 Lawler, Gregory, 150, 222, 254, 326, 343, 385 Ornstein–Uhlenbeck diffusion, 14, 61 Le Gall, J.-F., 117, 152, 187, 222, 254, 288, 383 outer boundary, 87, 326, 328, 344, 384 Lebesgue’s thorn, 250 outer measure, 100 Lévy, Paul, viii, 7, 9, 16, 22, 33, 50, 160, 212, 354 Lévy’s downward theorem, 354 packing dimension, 299 Lévy’s theorem packing number, 298 for local time, 212 Paley, Wiener, Zygmund’s theorem, 21 for the maximum process, 50 Paley–Wiener stochastic integral, 26 limsup fractal, 293 Paley–Zygmund inequality, 74, 91 Liouville’s theorem, 70, 253 Pemantle, Robin, 94, 117, 150, 384 local maximum, 39 percolation, 266 local time percolation interface, 337 at zero, 154 percolation limit set definition, 210 definition, 258 for planar Brownian motion, 385 generation dependent, 264 Lévy’s theorem, 212 perfect set, 48 Loewner evolution, 336 Perkins, Edwin, 34, 188, 287, 307, 326, 384 Loewner’s equation, 332, 344 permutation loop erasure, 334 finite, 19 loop-erased random walk, 333 Pitman’s theorem, 64, 152 lower envelope, 73 Pitman, Jim, 64, 118, 140, 152, 187, 222 Lyons’ theorem, 266 Plancherel’s theorem, 253 Lyons, Russell, 117, 265, 287, 325 Poincaré cone condition, 224 point of increase macroscopic, 1 global, 125 Makarov’s theorem, 254, 326 local, 123 Mandelbrot’s conjecture, 254 point recurrent, 72 Markov Poisson kernel, 85 strong property, 43 Poisson problem Markov process, 49 definition, 226 Markov property, 37 uniqueness, 251 Markov transition kernel, 49 Poisson’s formula, 85 Markov, Andrej, 63 Poisson, Siméon-Denis, 94 Martin kernel, 235 polar martingale, 53 for Brownian motion, 85 binary splitting, 127 for percolation, 259 discrete, 352 points, 47 reverse, 354 set, 234

Index 403

Portmanteau theorem, 347 stochastic process, 7 potential stopping time α-potential, 108 definition, 40 electrostatic, 84 for two-sided motion, 60 gravitational, 108 reverse, 60 Newtonian, 84 Strassen’s law, 150 potential kernel, 80, 235 subharmonic function, 65 product formula, 301 submartingale, 53 progressively measurable, 190 tail quadratic variation, 23, 195 0-1 law, 39 σ-algebra, 39 radial potential, 234 event, 39 random closed set, 263 Tanaka’s formula, 210 random field, 166 Taylor, S. James, 35, 109, 117, 144, 179, 188, 223, random walk, 1 258, 287, 290, 324 Ray’s theorem, 177 thick times, 289, 325 Ray–Knight theorem, 5, 170 thin times, 289, 325 record time, 106 transient, 72 recurrent, 72 trees neighbourhood, 72 terminology, 358 point, 72 Tricot, Claude, 326 reflected Brownian motion, 146 Trotter’s theorem, 165 reflection principle, 44, 70, 217 typical times, 20 regular point, 224 regularisation, 299 universal object, 1 removable set, 253 universality, 1 resolvent operator, 218 upcrossing excursion, 275 reverse Hölder, 102, 117 upper envelope, 118 reverse martingale, 354 Riemann mapping theorem, 202 value of a covering, 98 right-continuity, 42 value of a packing, 298 visible part of a set, 324 sample path properties, 7 scaling invariance, 4 Wald’s identity, see Wald’s lemma Schramm, Oded, 117, 150, 222 Wald’s lemma, 55 Schramm–Loewner evolution, see SLE first, 55 second arcsine law, 137, 215 second, 56 second moment method, 260 weak convergence, 252, 346 semimartingale, 64 Weierstrass function, 117 singular measure, 25 Werner, Wendelin, 150, 222, 384 Skorokhod embedding problem, 4 Wiener sausage, 116, 254 Skorokhod embedding theorem, 127 Wiener’s theorem, 9 SLE, 327 Wiener, Norbert, 9 chordal, 338, 339 Williams, David, 63, 146, 152, 187, 351 radial, 336, 339 winding number, 209 Smirnov, Stanislav, 337 spectrum, 291 Xiao, Yimin, 116, 289, 302 Spitzer’s law, 209 Spitzer, Frank, 94, 117, 209, 222, 254 Yor, Marc, 33, 187, 222 stable subordinator, 51, 252 stationary increments, 2 zero set stochastic integral Hausdorff dimension, 108 as martingale, 194 Hausdorff measure, 103 construction, 191 no isolated points, 48 continuous, 194 uncountable, 48 Fubini’s theorem, 221 zero-one law of Paley–Wiener, 26 for tail events, 39 up to finite time, 194 Galton–Watson, 286 stochastic integration, 6, 190 of Blumenthal, 38 stochastic Loewner evolution, see SLE of Hewitt–Savage, 19