Gaussove mere na Hilbertovih prostorih : delo diplomskega seminarja

Rems, Jan (author), Dragičević, Oliver (mentor)

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.

• Language: Slovenian
• Keywords: Hilbertov prostor, Gaussova mera, linearni operator, Gaussova slučajna spremenljivka, Fourierova transformiranka mere, funkcija belega šuma, Brownovo gibanje
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-108588
• UDC: 519.2
• COBISS.SI-ID: 18686297
• Publication date in RUL: 08.07.2019

REMS, Jan, 2019, Gaussove mere na Hilbertovih prostorih : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=108588

Secondary language

• Language: English
• Title: Gaussian measures in Hilbert spaces

Abstract

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.

• Keywords: Hilbert space, Gaussian measure, linear operator, Gaussian random variable, Fourier transform of measure, white noise function, Brownian motion