20.500.12556/RUL-108588 Gaussove mere na Hilbertovih prostorih delo diplomskega seminarja Gaussian measures in Hilbert spaces Na Hilbertovemu prostoru definiramo verjetnostno mero s pričakovano vrednostjo in kovariančnim operatorjem. Nato definiramo Gaussovo verjetnostno mero na prostoru realnih števil in jo za tem preko Fourierove transformiranke mere razširimo na Hilbertov prostor. S pomočjo produktne mere izračunamo nekaj Gaussovih integralov. Lastnosti Gaussovih mer prenesemo na Gaussove slučajne spremenljivke z vrednostmi v Hilbertovem prostoru in navedemo nekaj konkretnih primerov takšnih spremenljivk. Proučimo pogoje za neodvisnost le-teh in jih umestimo v prostor ekvivalenčnih razredov slučajnih spremenljivk. Nazadnje definiramo funkcijo belega šuma, ki elementu Hilbertovega prostora priredi slučajno spremenljivko, prav tako definirano na tem Hilbertovem prostoru. S pomočjo funkcije belega šuma in drugih pridobljenih rezultatov podamo konstrukcijo Brownovega gibanja. On a Hilbert space we define a probabilistic measure with expected value and covariance operator. Then Gaussian measure is defined on real line and later extended to Hilbert space by use of Fourier transform of measure. Notion of product measure helps us compute some integrals with respect to Gaussian measure. Properties of Gaussian measures are passed to Gaussian random variables with values in Hilbert space and some examples of these variables are presented. We investigate conditions for their indendence and put them in perspective of spaces of equivalence classes. At last we define a white noise function, which takes an element of Hilbert space and returns a random variable defined on a same Hilbert space. We use this result to construct a Brownian motion. Hilbertov prostor Gaussova mera linearni operator Gaussova slučajna spremenljivka Fourierova transformiranka mere funkcija belega šuma Brownovo gibanje Hilbert space Gaussian measure linear operator Gaussian random variable Fourier transform of measure white noise function Brownian motion true false false Slovenski jezik Angleški jezik Delo diplomskega seminarja/zaključno seminarsko delo/naloga 2019-07-08 15:58:15 2019-07-08 15:58:19 2024-05-30 11:10:14 0000-00-00 00:00:00 2019 0 0 0000-00-00 NiDoloceno NiDoloceno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.2 96124 18686297 RAZ_Rems_Jan_i2019.pdf RAZ_Rems_Jan_i2019.pdf 1 68EB9CA52FB7E7501BE2571D7869DA55 508d369dc35c153abb89d252e967ee270b5cd9beb8742e8043cf1847f85acef0 b9b12991-a1b6-11eb-a523-00155dcfd717 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=119886 Fakulteta za matematiko in fiziko 0 0 0