Index additive functional of random walk, 178, 284 adjoint operator, 314 aperiodic random walk, 134 Arzel´ a-Ascoli theorem, 311 Bessel identity, 299 Bessel-Clifford function of the second kind, 297 beta function, 307 Borel-Cantelli lemma, 82 extended Borel-Cantelli lemma, 82 Brownian motion, 25 Cameron-Martin formula, 127 Chapman-Kolmogorov equation, 172 Chung’s law of the iterated logarithm, 132 compound Poisson process, 292 convolution, 28, 300, 301 Cram´ er’s large deviation principle, 6 critical dimensions, 145 densely defined linear operator, 314 Dirac function, 25, 28 Dirichlet form, 95 Donsker-Varadhan’s large deviations, 128 entropy condition, 310 entropy method, 309 equicontinuity, 311 essential smoothness on R+, 11 essentially smooth function, 2 exponential moment generating function, 12 exponential Tauberian theorem, 24 exponential tightness, 7 Fenchel-Legendre transform, 2 Feynman-Kac formula, 91, 93 first entry formula, 137 Fourier inversion, 299, 300 Fourier transform, 299, 302 Fourier transformation, 95 Friedrichs’ extension theorem, 94, 315 G¨ artner-Ellis theorem on large deviations, 5 Gagliardo-Nirenberg inequality, 77, 86 good rate function, 2 Green’s function, 27, 297 — of random walk, 235, 236 partial — of random walk, 135 ground state solution, 86, 131 high moment asymptotics, 59, 86 hitting time, 137, 152 i.i.d. sequence, 6 increment functional of random walk, 178 infinitesimal generator, 93, 94 intersection local time, 25 p-multiple self —, 46 — of random walks, 177 double self —, 160 mutual —, 26, 27, 36 mutual — of random walks, 133, 144, 145, 187 renormalized p-multiple self—, 58, 132 renormalized self —, 48, 53, 111, 161, 177, 257 self —, 26 self — of random walk, 133 intersection of independent ranges, 133, 145, 177, 187 isometric linear operator, 299 Kolmogorov’s continuity theorem, 35, 310, 313 L´ evy process, 251 Lagrange multiplier, 305, 307 large deviation principle (LDP), 5 law of the iterated logarithm (LIL), 81, 121 — for Brownian motions, 83 Le Gall’s moment identity, 33 local time, 36, 37, 102, 139, 152 logarithmic moment generating function, 1, 24 lower semi-continuity, 2 Markov process, 56 331

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