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 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 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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