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.

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.

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.

Brownian motion has met growing interest in mathematics, physics and particularly in finance since it was introduced in the beginning of the twentieth century. Stochastic processes generalizing Brownian motion have influenced many research fields theoretically and practically. Moreover, along with more refined techniques in measure theory and functional analysis more stochastic processes were constructed and studied. Lévy processes, with Brownian motionas a special case, have been of major interest in the recent decades. In addition, Lévy processes include a number of other important processes as special cases like Poisson processes and subordinators. They are also related to stable processes.

In this thesis we generalize a result by S. Chandrasekhar [2] and Edward Nelson who gave a detailed proof of this result in his book in 1967 [12]. In Nelson’s first result standard Ornstein-Uhlenbeck processes are studied. Physically this describes free particles performing a random and irregular movement in water caused by collisions with the water molecules. In a further step he introduces a nonlinear drift in the position variable, i.e. he studies the case when these particles are exposed to an external field of force in physical terms.

In this report, we aim to generalize the result of Edward Nelson to the case of α-stable Lévy processes. In other words we replace the driving noise of a standard Ornstein-Uhlenbeck process by an α-stable Lévy noise and introduce a scaling parameter uniformly in front of all vector fields in the cotangent space, even in front of the noise. This corresponds to time being sent to infinity. With Chandrasekhar’s and Nelson’s choice of the diffusion constant the stationary state of the velocity process (which is approached as time tends to infinity) is the Boltzmann distribution of statistical mechanics.The scaling limits we obtain in the absence and presence of a nonlinear drift term by using the scaling property of the characteristic functions and time change, can be extended to other types of processes rather than α-stable Lévy processes.

In future, we will consider to generalize this one dimensional result to Euclidean space of arbitrary finite dimension. A challenging task is to consider the geodesic flow on the cotangent bundle of a Riemannian manifold with scaled drift and scaled Lévy noise. Geometrically the Ornstein-Uhlenbeck process is defined on the tangent bundle of the real line and the driving Lévy noise is defined on the cotangent space.

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.

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.

This study focuses on students' way of reasoning about a proof in mathematics. The experiences of teaching students in the beginning of their studies at universities show that students have an obstacle in using deductive methods. The students' activity was designed specifically to investigate their deductive ability and to see if they can develop their way of reasoning. The group activities and interviews follow the students from the beginning where they, with great enthusiasm, begin colouring maps as a first sketch to a complete proof. The well-known statement to prove is chosen from a field in mathematics that the students are unfamiliar with, namely graph theory. More precisely it concerns the number of possible colourings of maps. Some university students have problems with constructing proofs, but in many cases the teacher can help them to reach a deductive reasoning.