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Nelson-type Limit for a Particular Class of Lévy Processes
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
Linnéuniversitetet, Fakultetsnämnden för naturvetenskap och teknik, Institutionen för datavetenskap, fysik och matematik, DFM.
Department of Statistics, University of Warwick, CV4 7AL, UK.
2010 (Engelska)Ingår i: AIP Conference Proceedings; 1232 / [ed] Andrei Yu. Khrennikov, AIP , 2010, Vol. 1232, s. 189-193Konferensbidrag, Publicerat paper (Övrigt vetenskapligt)
Abstract [en]

Brownian motion has been constructed in different ways. Einstein was the most outstanding physicists involved in its construction. From a physical point of view a dynamical theory of Brownian motion was favorable. The Ornstein-Uhlenbeck process models such a dynamical theory and E. Nelson amongst others derived Brownian motion from Ornstein-Uhlenbeck theory via a scaling limit. In this paper we extend the scaling result to α-stable Lévy processes.

Ort, förlag, år, upplaga, sidor
AIP , 2010. Vol. 1232, s. 189-193
Serie
AIP Conference Proceedings, ISSN 0094-243X ; 1232
Forskningsämne
Naturvetenskap, Matematik
Identifikatorer
Scopus ID: 2-s2.0-77955376933ISBN: 978-0-7354-0777-0 (tryckt)OAI: oai:DiVA.org:lnu-5970DiVA, id: diva2:323159
Konferens
Quantum Theory: Reconsideration of Foundations - 5, Växjö (Sweden), 14–18 June 2009
Ingår i avhandling
1. A Differentiable Approach to Stochastic Differential Equations: the Smoluchowski Limit Revisited
Öppna denna publikation i ny flik eller fönster >>A Differentiable Approach to Stochastic Differential Equations: the Smoluchowski Limit Revisited
2012 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
Abstract [en]

In this thesis we generalize results by Smoluchowski [43], Chandrasekhar[6], Kramers, and Nelson [30]. Their aim is to construct Brownian motion as a limit of stochastic processes with differentiable sample paths by exploiting a scaling limit which is a particular type of averaging studied by Papanicolao [35]. Their construction of Brownian motion differs from the one given by Einstein since it constitutes a dynamical theory of Brownian motion. Nelson sets off by studying scaled standard Ornstein-Uhlenbeck processes. Physically these describe classical point particles subject to a deterministic friction and an external random force of White Noise type, which models perpetuous collisions with surrounding(water) molecules. Nelson also studies the case when the particles are subject to an additional deterministic nonlinear force. The present thesis generalizes the work of Chandrasekhar in that it deals with finite dimensional α-stable Lévy processes with 0 < α < 2, and Fractional Brownian motion as driving noises and mathematical techniques like deterministic time change and a Girsanov theorem. We consider uniform convergence almost everywhere and in $L^2$-sense. In order to pursue the limit we multiply all vector fields in the cotangent space by the scaling parameter including the noise. For α-stable Lévy processes this correspondsto scaling the process in the tangent space, $v_{t}$, $t\geq 0$, according to $\beta v_{t}=v_{\beta^{\alpha}t}$. Sending β to infinity means sending time to infinity. In doing so the noise evolves with a different speed in time compared to the component processes. For α≠2, α-stable Lévy processes are of pure jump type, therefore the approximation by processes having continuous sample paths constitutes a valuable mathematical tool. α-stable Lévy processes exceed the class studied by Zhang [46]. In another publication related to this thesis we elaborate on including a mean-field term into the globally Lipschitz continuous nonlinear part of the drift while the noise is Brownian motion, whereas Narita [28] studied a linear dissipation containing a mean-field term. Also the classical McKean-Vlasov model is linear in the mean-field. In a result not included in this thesis the scaling result of Narita [29], which concerns another scaling limit of the tangent space process (velocity) towards a stationary distribution, is generalized to α-stable Lévy processes. The stationary distribution derived by Narita is related to the Boltzmann distribution. In the last part of this thesis we study Fractional Brownian motion with a focus on deriving a scaling limit of Smoluchowski-Kramers type. Since Fractional Brownian motion is no semimartingale the underlying theory of stochastic differential equations is rather involved. We choose to use a Girsanov theorem to approach the scaling limit since the exponent in the Girsanov denvsity does not contain the scaling parameter explicitly. We prove that the Girsanov theorem holds with a linear growth condition alone on the drift for 0 < H < 1, where H is the Hurst parameterof the Fractional Brownian motion.

Ort, förlag, år, upplaga, sidor
Växjö, Kalmar: Linnaeus University Press, 2012. s. 23
Serie
Linnaeus University Dissertations ; 103
Nyckelord
α-stable Lévy noise, Fractional Brownian motion, Girsanov theorem, Mean-field model, Nonlinear stochastic oscillator, Ornstein-Uhlenbeck process, Scaling limit, Second order Itô equation, Time change.
Nationell ämneskategori
Matematik Sannolikhetsteori och statistik
Forskningsämne
Matematik, Matematik
Identifikatorer
urn:nbn:se:lnu:diva-22233 (URN)978-91-86983-85-7 (ISBN)
Disputation
2012-11-22, D1136, vejdes plats 7, Växjö, 10:15 (Engelska)

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