On some Inferences of Lévy Distribution

The levy distribution is one of the three distributions that has probability density function in simple closed form. This distribution is used in modeling stock prices. In this paper, we present some properties of this distribution. Based on the basic properties some characterizations of this distribution are given.

Simulations of Lévy walk

We simulate stable distributions to study the ideal movement pattern for the spread of a virus using autonomous carrier. We observe Lévy walks to be the most ideal way to spread and further study how the parameters in Lévy distribution affects the spread.

Lymphoidal chemokine CCL19 promoted the heterogeneity of the breast tumor cell motility within a 3D microenvironment revealed by a Lévy distribution analysis

Tumor cell heterogeneity, either at the genotypic or the phenotypic level, is a hallmark of cancer. Tumor cells exhibit large variations, even among cells derived from the same origin, including cell morphology, speed and motility type. However, current work for quantifying tumor cell behavior is largely population based and does not address the question of cell heterogeneity. In this article, we utilize Lévy distribution analysis, a method known in both social and physical sciences for quantifying rare events, to characterize the heterogeneity of tumor cell motility. Specifically, we studied the breast tumor cell (MDA-MB-231 cell line) velocity statistics when the cells were subject to well-defined lymphoid chemokine (CCL19) gradients using a microfluidic platform. Experimental results showed that the tail end of the velocity distribution of breast tumor cell was well described by a Lévy function. The measured Lévy exponent revealed that cell motility was more heterogeneous when CCL19 concentration was near the dynamic kinetic binding constant to its corresponding receptor CCR7. This work highlighted the importance of tumor microenvironment in modulating tumor cell heterogeneity and invasion.

Econophysics and Fractional Calculus: Einstein’s Evolution Equation, the Fractal Market Hypothesis, Trend Analysis and Future Price Prediction

This paper examines a range of results that can be derived from Einstein’s evolution equation focusing on the effect of introducing a Lévy distribution into the evolution equation. In this context, we examine the derivation (derived exclusively from the evolution equation) of the classical and fractional diffusion equations, the classical and generalised Kolmogorov–Feller equations, the evolution of self-affine stochastic fields through the fractional diffusion equation, the fractional Poisson equation (for the time independent case), and, a derivation of the Lyapunov exponent and volatility. In this way, we provide a collection of results (which includes the derivation of certain fractional partial differential equations) that are fundamental to the stochastic modelling associated with elastic scattering problems obtained under a unifying theme, i.e., Einstein’s evolution equation. This includes an analysis of stochastic fields governed by a symmetric (zero-mean) Gaussian distribution, a Lévy distribution characterised by the Lévy index γ ∈ [ 0 , 2 ] and the derivation of two impulse response functions for each case. The relationship between non-Gaussian distributions and fractional calculus is examined and applications to financial forecasting under the fractal market hypothesis considered, the reader being provided with example software functions (written in MATLAB) so that the results presented may be reproduced and/or further investigated.