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