Generalized stochastic processes with applications in equation solving / Uopšteni stohastički procesi sa primenama u rešavanju jednačina

Description

<p>In this dissertation stochastic processes are regarded in the framework of Colombeau-type algebras of generalized functions. Such processes are called Colombeau stochastic processes.The notion of point values of Colombeau stochastic processes in compactly supported generalized points is established. The Colombeau algebra of compactly supported generalized constants is endowed with the topology generated by sharp open balls. The measurability of the corresponding random variables with values in the Colombeau algebra of compactly supported generalized constants is shown.<br />The generalized correlation function and the generalized characteristic function of Colombeau stochastic processes are introduced and their properties are investigated. It is shown that the characteristic function of classical stochastic processes can be embedded into the space of generalized characteristic functions. Examples of generalized characteristic function related to gaussian Colombeau stochastic<br />processes are given. The structural representation of the generalized correlation function which is supported on the diagonal is given. Colombeau stochastic processes with independent values are introduced. Strictly stationary and weakly stationary Colombeau stochastic processes are studied. Colombeau stochastic processes with stationary increments are characterized via their stationarity of the gradient of the process.Gaussian stationary solutions are analyzed for linear stochastic partial differential equations with generalized constant coefficients in the framework of Colombeau stochastic processes.</p> / <p>U disertaciji se stohastički procesi posmatraju u okviru Kolomboove algebre uop&scaron;tenih funkcija. Takve procese nazivamo Kolomboovi stohastički procesi. Pojam vrednosti Kolomboovog stohastičkog procesa u tačkama sa kompaktnim nosačem je uveden. Dokazana je merljivost odgovarajuće slučajne promenljive sa vrednostima u Kolomboovoj algebri uop&scaron;tenih konstanti sa kompaktnim nosačem,&nbsp; snabdevenom topologijom generisanom o&scaron;trim otvorenim loptama. Uop&scaron;tena korelacijska funkcija i uop&scaron;tena karakteristična funkcija Kolomboovog stohastičkog procesa su definisane i njihove osobine su izučavane. Pokazano je da&nbsp; se karakteristična funkcija klasičnog stohastičkog procesa može potopiti u prostor uop&scaron;tenih karakterističnih funkcija. Dati su primeri uop&scaron;tenih karakterističnih funkcija&nbsp; gausovskih Kolomboovih stohastičkih procesa. Data je strukturna reprezentacija uop&scaron;tene korelacijske funkcije sa nosačem na dijagonali. Kolomboovi stohastički procesi sa nezavisnim vrednostima su predstavljeni. Izučavani su strogo stacionarni i&nbsp; slabo stacionarni Kolomboovi stohastički procesi. Kolomboovi stohastički procesi sa stacionarnim prira&scaron;tajima su okarakterisani preko stacionarnosti gradijenta procesa. Gausovska stacionarna re&scaron;enja za linearnu stohastičku parcijalnu diferencijalnu jednačinu sa uop&scaron;tenim konstantnim koeficijentima su analizirana u okvirima Kolomboovih stohastičkih procesa.</p>

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1. https://www.cris.uns.ac.rs/DownloadFileServlet/Disertacija154928516102933.pdf?controlNumber=(BISIS)110199&fileName=154928516102933.pdf&id=12568&source=NDLTD&language=en
2. https://www.cris.uns.ac.rs/record.jsf?recordId=110199&source=NDLTD&language=en

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Additional Fields

Identifer	oai:union.ndltd.org:uns.ac.rs/oai:CRISUNS:(BISIS)110199
Date	10 May 2019
Creators	Gordić Snežana
Contributors	Pilipović Stevan, Oberguggenberger Michael, Seleši Dora, Rajter-Ćirić Danijela, Oparnica Ljubica
Publisher	Univerzitet u Novom Sadu, Prirodno-matematički fakultet u Novom Sadu, University of Novi Sad, Faculty of Sciences at Novi Sad
Source Sets	University of Novi Sad
Language	English
Detected Language	English
Type	PhD thesis