Related papers: Dynamics of Mandelbrot Cascades
We investigate piecewise deterministic Markov processes (PDMP), where the deterministic dynamics follows a scalar conservation law and random jumps in the system are characterized by changes in the flux function. We show under which…
Macdonald processes are probability measures on sequences of partitions defined in terms of nonnegative specializations of the Macdonald symmetric functions and two Macdonald parameters q,t in [0,1). We prove several results about these…
We study an extended dynamical system on the non-negative real line with piecewise linear non-uniformly expanding local dynamics. With a uniformly distributed initial state, the distribution of successive states coincides with that of a…
Dynamical systems describe the changes in processes that arise naturally from their underlying physical principles, such as the laws of motion or the conservation of mass, energy or momentum. These models facilitate a causal explanation for…
Cascades on random networks are typically analyzed by assuming they map onto percolation processes and then are solved using generating function formulations. This approach assumes that the network is infinite and weakly connected, yet…
Multifractal analysis of multiplicative random cascades is revisited within the framework of {\em mixed asymptotics}. In this new framework, statistics are estimated over a sample which size increases as the resolution scale (or the…
Under certain mild conditions, limit theorems for additive functionals of some $d$-dimensional self-similar Gaussian processes are obtained. These limit theorems work for general Gaussian processes including fractional Brownian motions,…
The mean-squared displacement (MSD) is an averaged quantity widely used to assess anomalous diffusion. In many cases, such as molecular motors with finite processivity, dynamics of the system of interest produce trajectories of varying…
In this work, cascading transmission line failures are studied through a dynamical model of the power system operating under fixed conditions. The power grid is modeled as a stochastic dynamical system where first-principles…
We introduce a path sampling method for obtaining statistical properties of an arbitrary stochastic dynamics. The method works by decomposing a trajectory in time, estimating the probability of satisfying a progress constraint, modifying…
Despite their deterministic nature, dynamical systems often exhibit seemingly random behaviour. Consequently, a dynamical system is usually represented by a probabilistic model of which the unknown parameters must be estimated using…
Bayesian learning using Gaussian processes provides a foundational framework for making decisions in a manner that balances what is known with what could be learned by gathering data. In this dissertation, we develop techniques for…
In this work we consider time series with a finite number of discrete point changes. We assume that the data in each segment follows a different probability density functions (pdf). We focus on the case where the data in all segments are…
Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such…
We consider the winding number of planar stationary Gaussian processes defined on the line. Under mild conditions, we obtain the asymptotic variance and the Central Limit Theorem for the winding number as the time horizon tends to infinity.…
We consider a finite sequence of random points in a finite domain of a finite-dimensional Euclidean space. The points are sequentially allocated in the domain according to a model of cooperative sequential adsorption. The main peculiarity…
We consider the dynamics of a droplet on a vibrating fluid bath. This hydrodynamic quantum analog system is shown to elicit the canonical behavior of damped-driven systems, including a period doubling route to chaos. By approximating the…
Many biological phenomena such as locomotion, circadian cycles, and breathing are rhythmic in nature and can be modeled as rhythmic dynamical systems. Dynamical systems modeling often involves neglecting certain characteristics of a…
We define a large class of continuous time multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined…
Simple models of irreversible dynamical processes such as Bootstrap Percolation have been successfully applied to describe cascade processes in a large variety of different contexts. However, the problem of analyzing non-typical…