2012-10-29
Disputation i matematik: Haidar Al-Talibi
Torsdagen den 22 november disputerar Haidar Al-Talibi i ämnet matematik.
Torsdagen den 22 november 2012 kl 10.15 i sal D1136 på Linnéuniversitetet i Växjö lägger Haidar Al-Talibi fram sin doktorsavhandling i matematik. Avhandlingen bär titeln "A Differentiable Approach to Stochastic Differential Equations – The Smoluchowski Limit Revisited". Opponent är professor Yaozhong Hu, University of Kansas, USA.
Spikningen sker onsdagen den 31 oktober 2012 kl 13.00 på Universitetsbiblioteket i Växjö.
Abstract
Please note that the super- and subscripts in the text below do not appear correctly.
The present thesis generalizes the work of Chandrasekhar (1943) 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 L2-sense. In order to pursue the limit we multiply all vector fields in the cotangent space by the scaling parameter β including the noise. For α-stable Lévy processes this corresponds to scaling the process in the tangent space, vt , t ≥ 0, according to βvt = vβαt. 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.
In another publication related to this thesis we elaborate on including a mean field term into the globally Lipshitz continuous nonlinear part of the drift while the noise is Brownian motion, whereas Narita (1991) studied a linear dissipation containing a mean field term. Also the classical McKean-Vlasov model is linear in the mean field.
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 density 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 parameter of the Fractional Brownian motion.