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.
Friday, June 11, 2021 (9AM CT US)
Dr Alexei Daletskii "Stochastic Camassa-Holm equation with convection type noise: local existence and uniqueness"
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.
Friday, May 28, 2021 (9AM CT US)
Vitali Vougalter "On the solvability of some systems of
integro-differential equations with anomalous diffusion in higher
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.
Friday, May 14, 2021 (9AM CT US)
Michele Ricciardi "VISCOSITY SOLUTIONS FOR PARABOLIC TRANSPORT
EQUATIONS (WITH APPLICATIONS TO RTDS)"
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.
Friday, Apr 16, 2021 (9AM CT US)
Mark Edelman "Cycles in discrete fractional (with power-law memory) systems"
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
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
process is Markov.
Friday, Mar 12, 2021 (9AM CT US)
Dimitri Volchenkov "Discrete Time Markov Chains with Random Transition Times –Fractional Markov Chains"
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.