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A Differentiable Approach to Stochastic Differential Equations: the Smoluchowski Limit Revisited
Linnaeus University, Faculty of Science and Engineering, School of Computer Science, Physics and Mathematics.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
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 -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, , , according to . 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.

Place, publisher, year, edition, pages
Växjö, Kalmar: Linnaeus University Press, 2012. , 23 p.
Series
Linnaeus University Dissertations, 103
Keyword [en]
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.
National Category
Mathematics Probability Theory and Statistics
Research subject
Mathematics, Mathematics
Identifiers
URN: urn:nbn:se:lnu:diva-22233ISBN: 978-91-86983-85-7 (print)OAI: oai:DiVA.org:lnu-22233DiVA: diva2:563341
Public defence
2012-11-22, D1136, vejdes plats 7, Växjö, 10:15 (English)
Opponent
Supervisors
Available from: 2012-10-31 Created: 2012-10-29 Last updated: 2012-10-31Bibliographically approved
List of papers
1. Differentiable Approximation of Diffusion Equations Driven by α-Stable Lévy Noise
Open this publication in new window or tab >>Differentiable Approximation of Diffusion Equations Driven by α-Stable Lévy Noise
2013 (English)In: Brazilian Journal of Probability and Statistics, ISSN 0103-0752, E-ISSN 2317-6199, Vol. 27, no 4, 544-552 p.Article in journal (Refereed) Published
Abstract [en]

Edward Nelson derived Brownian motion from Ornstein-Uhlenbeck theory by a scaling limit. Previously we extended the scaling limit to an Ornstein-Uhlenbeck process driven by an α-stable Lévy process. In this paper we extend the scaling result to α-stable Lévy processes in the presence of a nonlinear drift, an external field of force in physical terms.

Keyword
Ornstein-Uhlenbeck process, α-stable Lévy noise, scaling limits
National Category
Probability Theory and Statistics
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-11415 (URN)10.1214/11-BJPS180 (DOI)000325443900007 ()
Available from: 2011-04-14 Created: 2011-04-14 Last updated: 2016-09-22Bibliographically approved
2. Differentiable Approximation by Solutions of Newton Equations Driven by Fractional Brownian Motion.
Open this publication in new window or tab >>Differentiable Approximation by Solutions of Newton Equations Driven by Fractional Brownian Motion.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We derive a Smoluchowski-Kramers type scaling limit for second order stochastic differential equations driven by Fractional Brownian motion.We show a Girsanov theorem for the solution processes with respect to corresponding Fractional Ornstein-Uhlenbeck processes which are Gaussian. This reveals existence of weak solutions as well as a weak scaling limit. Subsequently the results are strengthened.

Keyword
Fractional Ornstein-Uhlenbeck process, Fractional Brownian motion, Second order stochastic differential equation, scaling limit, Smoluchowski-Kramers limit
National Category
Probability Theory and Statistics
Research subject
Mathematics, Mathematics
Identifiers
urn:nbn:se:lnu:diva-16560 (URN)
Available from: 2012-01-04 Created: 2012-01-04 Last updated: 2012-10-31Bibliographically approved
3. Nelson-type Limit for a Particular Class of Lévy Processes
Open this publication in new window or tab >>Nelson-type Limit for a Particular Class of Lévy Processes
2010 (English)In: AIP Conference Proceedings; 1232 / [ed] Andrei Yu. Khrennikov, AIP , 2010, Vol. 1232, 189-193 p.Conference paper, (Other academic)
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.

Place, publisher, year, edition, pages
AIP, 2010
Series
AIP Conference Proceedings, ISSN 0094-243X ; 1232
Research subject
Natural Science, Mathematics
Identifiers
urn:nbn:se:lnu:diva-5970 (URN)10.1063/1.3431487 (DOI)978-0-7354-0777-0 (ISBN)
Conference
Quantum Theory: Reconsideration of Foundations - 5, Växjö (Sweden), 14–18 June 2009
Available from: 2010-06-09 Created: 2010-06-09 Last updated: 2012-10-31Bibliographically approved
4. Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift
Open this publication in new window or tab >>Smoluchowski-Kramers Limit for a System Subject to a Mean-Field Drift
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We establish a scaling limit for autonomous stochastic Newton equations, the solutions are often called nonlinear stochastic oscillators,where the nonlinear drift includes a mean field term of Mckean type and the driving noise is Gaussian. Uniform convergence in  sense is achieved by applying -type estimates and the Gronwall Theorem.The approximation is also called Smoluchowski-Kramers limit and is a particular averaging technique studied by Papanicolaou. It reveals an approximation of diffusions with a mean-field contribution in the drift by diffusions with differentiable trajectories.

Keyword
Averaging, McKean equation, Mean-field model, Nonlinear stochastic oscillator, Second order Itô equation, Smoluchowski- Kramers limit
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:lnu:diva-19216 (URN)
Available from: 2012-05-31 Created: 2012-05-31 Last updated: 2013-08-15Bibliographically approved

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