Random Time, Memory, and Fractional Dynamics Seminar

Individual-based models are used to study complex phenomena in many fields of science. While simulating agent-based models is often straightforward, predicting their behaviour mathematically has remained a key challenge. Recently developed mathematical methods allow the prediction of the emerging spatial patterns for a general class of agent-based models, whereas the prediction of spatio-temporal pattern has been thus far achieved only for special cases. We present a general technique that allows deriving the spatio-temporal (pair) correlation structure for a general class of individual-based models. To do so, we define an auxiliary model, in which each agent type of the primary model expands to three types, called the original, the past and the new agents. In this way, the auxiliary model keeps track of both the initial and current state of the primary model, and hence the spatio-temporal correlations of the primary model can be derived from the spatial correlations of the auxiliary model. We illustrate also the agreement between analytical predictions and agent-based simulations using two example models from theoretical ecology.

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first-order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato’s operator theory methods.

The work deals with the studies of the existence of solutions of a system of integro-differential equations in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in R^d, d=4,5. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.

Nonlocal generators are related with pure jump Markov processes, in particular, with compound Poisson ones. We analyze properties of such generators and their characteristics. Potential perturbations of these generators lead to the notion of nonlocal Schroedinger operators (NLSO). We will discuss spectral properties of NLSP and first of all the ground state problem. The latter has an important meaning in applications.

In this talk we investigate the well-posedness in Rd of parabolic transport equations $u_t = b \cdot \nabla u + cu + f$. The classical viscosity theory, developed for elliptic equation of first order and parabolic equation of second order, can be applied in this framework, and we are able to obtain existence, uniqueness, Lipschitz and semiconcave estimates for the solution $u$. Then we use these results in order to study the behaviour of the solution under a random time change, where the random process is an inverse subordinator of a compound Poisson process.

We study the effect of subordination to the solution of a model of spatial ecology in terms of the evolution density. The asymptotic behavior of the subordinated solution for different rates spatial propagation is studied. The difference between subordinated solutions to non-linear equations with classical time derivative and solutions to non-linear equation with fractional time derivative is discussed. We show how the intermittency property may appear as the result of the random time change.

In this talk we will introduce a generalization of Brownian motion which leads to a class of in general non-Gaussian processes. We present results obtained for this process and give a brief overview about its properties. In addition, we give a representation using grey Ornstein-Uhlenbeck processes. An ergodicity breaking parameter shows why the process is challenging from the view of numerical simulations. We conclude with a short overview of generalized scaling operators and transformation groups in this framework.

There I discuss a generalization of the notion of natural numbers and related topics. It corresponds to the extension of population descriptions from characterization by number of individuals to spatially distributed populations. In this way does appear continuous combinatorics, difference calculus in the continuum etc. Actually such extension means the transition from 0D statistical physics to the continuous models.

Presence of the power-law memory is a significant feature of many natural (biological, physical, etc.) and social systems. Continuous and discrete fractional calculus is the instrument to describe the behavior of systems with the power-law memory. Existence of chaotic solutions is an intrinsic property of nonlinear dynamics (regular and fractional). Behavior of fractional systems can be very different from the behavior of the corresponding systems with no memory. Finding periodic points is essential for understanding regular and chaotic dynamics. Fractional systems don’t have periodic points except fixed points. Instead, they have asymptotically periodic points (sinks). There have been no reported results (formulae) which would allow calculations of asymptotically periodic points of nonlinear fractional systems so far. In this presentation we derive the equations that allow calculations of the coordinates of the asymptotically periodic sinks.

The concept of random times does appear in several real world models in the contrast to the Newton time notion usual in classical mechanics. Our aim is to show how a random time will change the behavior of considered systems. We consider two classes of dynamics. At first, random time Markov processes will be analyzed. Secondly, we study random time deterministic dynamical systems which are (in certain sense) special cases of Markov evolution.

In this talk we present Green measures of certain classes of Markov processes. In particular Brownian motion and processes with jump generators with different tails. The Green measures are represented as a sum of a singular and a regular part given in terms of the jump generator. The main technical issue is to obtain a bound for the regular part. Several classes of jump kernels will be analysed. In addition we also investigate Green measure for time changed Markov processes where time is given by the inverse of a subordinator (most interesting case). Here the notion of renormalised Green measure is needed to obtain a well defined object.

The asymptotic properties of a symmetric random walk in a high contrast periodic medium on the lattice are considered.We show that under proper diffusive scaling the random walk exhibits a non-standard limit behaviour. In addition to the coordinate of the random walk in $\mathbb Z^d$ we introduce an extra variable that characterizes the position of the random walk in the period and show that this two-component process converges in law to a limit Markov process. The components of the limit process are mutually coupled, thus we cannot expect that the limit behaviour of the coordinate process is Markov.

The exact amounts of predictable and unpredictable information in fractional coin flipping (at random times) defined as a binomial power series of the "integer" flipping are reported. Due to strong coupling between the tossing outcomes at different times, the side repeating probabilities assumed to be independent for “integer” flipping get entangled with one another for fractional flipping. The predictable and unpredictable information components vary smoothly with the fractional order parameter.