332 INDEX irreducible —, 84 symmetric —, 85 transition probability of —, 56 Minkowski functional, 8 moderate deviation, 222, 248 modification of stochastic processes, 35 continuous modification, 35, 52 modified Bessel equation, 297 multinomial inequality, 181 non-negative operator, 93, 98 occupation measure, 36 Orlicz norm, 309 Orlicz space, 309 Parseval identity, 299, 301, 302 period of random walk, 134 periodic function, 134, 301, 302 Plancherel-Parseval theorem, 54, 302 Poisson process, 22 polymer models, 111 positively balanced set, 8 probability of no return, 138 projection operator, 315 Prokhorov criterion, 310 Radon measure, 44 random walk, 133 random walk in random scenery, 174, 292 range of random walk, 133, 160, 177 rapidly decreasing function, 93, 300, 301 rate function, 2 recurrence, 138 renormalization, 48 resolution of identity, 97, 315, 317 resolvent approximation, 136 resolvent equation, 306 resolvent random walk, 136 reverse Markov inequality, 57 Schwartz space, 94, 300 self-adjoint operator, 61, 209, 314 function of self-adjoint operator, 318 self-attracting polymer, 26 self-repelling polymer, 26 semi-bounded operator lower —, 315 upper —, 94, 315 semi-group, 92 simple random walk, 134, 145 small ball probability, 23 Sobolev inequality, 303 Sobolev space, 303 spectral decomposition, 314 spectral integral, 96, 316 spectral integral representation, 61, 210, 314, 317 spectral measure, 97, 315 spherically symmetric function, 131 steep function, 3 sub-additive functional of random walk, 178 sub-additive sequence, 19 deterministic —, 19 sub-additive stochastic process, 21 sub-additivity, 1, 91, 117 sub-critical dimensions, 145 super-critical dimensions, 145, 173 symmetric operator, 92, 94, 208, 314 thick point, 173 topological dual space, 107 transience, 138 triangular approximation, 49, 50, 161, 192, 258 uniform exponential integrability, 18 uniform tightness, 8, 139, 310 Varadhan’s integral lemma, 6 Wiener sausage, 249, 291, 293, 294 Young function, 309

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