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