Behavior Of Brownian Motion Research Articles

Brownian motion from deterministic dynamics

Certain deterministic dynamical systems exhibit a transition from non-Gaussian chaotic to Brownian motion behavior when a suitable scaling limit is applied. For a simple hyperbolic model system we analyze the structure of the corresponding strange attractor, determine the invariant measure and elucidate the transition scenario leading to a Gaussian stochastic process.

A strong invariance principle for reverse martingales

In a previous work (Scott and Huggins [5]) an embedding of reverse martingales in Brownian motion was obtained and used to give a law of the iterated logarithm for reverse martingales using properties of Brownian motion near the origin. This iterated logarithm law complements the iterated logarithm law for martingales in the same manner as results concerning the behaviour of Brownian motion as t-*0 complement those concerning the behaviour of Brownian motion as t-* ~. This becomes clearer from an examination of the embedding of doubly infinite martingales in Brownian motion given as Theorem 3 of Scott and Huggins [5]. Here we sharpen the underlying invariance principle of Scott and Huggins [5] and this result complements the corresponding martingale result of Jain, Jogdeo and Stout [3]. We then give, as an application of this invariance principle, integral tests for upper and lower functions of reverse martingales which again complement the result of Jain, Jogdeo and Stout. As we quite often follow the proofs of Jain, Jogdeo and Stout fairly closely with the appropriate changes necessary for the reverse martingale case in some places only a sketch of the proof is needed.

Stochastic theory for classical and quantum mechanical systems

We formulate from first principles a theory of stochastic processes in configuration space. The fundamental equations of the theory are an equation of motion which generalizes Newton's second law and an equation which expresses the condition of conservation of matter. Two types of stochastic motion are possible, both described by the same general equations, but leading in one case to classical Brownian motion behavior and in the other to quantum mechanical behavior. The Schrodinger equation, which is derived here with no further assumption, is thus shown to describe a specific stochastic process. It is explicitly shown that only in the quantum mechanical process does the superposition of probability amplitudes give rise to interference phenomena; moreover, the presence of dissipative forces in the Brownian motion equations invalidates the superposition principle. At no point are any special assumptions made concerning the physical nature of the underlying stochastic medium, although some suggestions are discussed in the last section.

On some Asymptotic Properties Concerning Homogeneous Differential Processes

About the behaviour of brownian motion at time point ∞ there are many results by P. Levy and A. Khintchine etc. The method of W. Feller is applicable to a similar discussion about a homogeneous differential process. In this paper we shall study, applying his method, the properties of a homogeneous differential process.

On Some Asymptotic Properties Concerning Brownian Motion

About the behavior of brownian motion at time point oo, there are many results by P. Lévy, A. Khintchine etc. P. Lévy cited a theorem by A. Kolmogoroff as the most precise result in his famous book “Processus stochastiques et mouvement brownian” without proof. In this paper we shall prove this theorem, using the similar result about the random sequence by W. Feller, and then, applying the theorem of projective invariance by P. Lévy, we shall find also the behavior of brownian motion at time point 0 from the above theorem.

Anomalous diffusion in comb-shaped domains and graphs

In this paper we study the asymptotic behavior of Brownian motion in both comb-shaped planar domains, and comb-shaped graphs. We show convergence to a limiting process when both the spacing between the teeth \emph{and} the width of the teeth vanish at the same rate. The limiting process exhibits an anomalous diffusive behavior and can be described as a Brownian motion time-changed by the local time of an independent sticky Brownian motion. In the two dimensional setting the main technical step is an oscillation estimate for a Neumann problem, which we prove here using a probabilistic argument. In the one dimensional setting we provide both a direct SDE proof, and a proof using the trapped Brownian motion framework in Ben Arous \etal (Ann. Probab. '15).