Gaussove mere na Hilbertovih prostorih

delo diplomskega seminarja

Jan Rems (Author), Oliver Dragičević (Mentor)

Abstract

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.

Keywords

matematika;Hilbertovi prostori;Gaussova mera;linearni operatorji;Gaussove slučajne spremenljivke;Fourierova transformiranka mere;funkcija belega šuma;Brownovo gibanje;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.11 - Undergraduate Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[J. Rems]
UDC:	519.2
COBISS:	18686297
Views:	1443
Downloads:	344
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Gaussian measures in Hilbert spaces
Secondary 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.
Secondary keywords:	mathematics;Hilbert spaces;Gaussian measure;linear operators;Gaussian random variables;Fourier transform of measure;white noise function;Brownian motion;
Type (COBISS):	Final seminar paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Finančna matematika - 1. stopnja
Pages:	38 str.
ID:	11187005