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Marta Tyran-Kaminska

Publications and source records attributed to Marta Tyran-Kaminska.

16 recordsLinked to original sources
arXiv2024

Temporal Entropy Evolution in Stochastic and Delayed Dynamics

We review the behaviour of the Gibbs' and conditional entropies in deterministic and stochastic systems and continue to a formulation appropriate for a stochastically perturbed system with delayed dynamics. The underlying question driving these investigations: ``Is the origin of the universally observed unidirectionality of time in our universe connected to the behaviour of entropy?'' We focus on temporal entropic behaviour with a review of previous results in deterministic and stochastic systems. Our emphasis is on the temporal behaviour of the Gibbs' and conditional entropies as these entropies give equilibrium results in concordance with experimental findings. In invertible deterministic systems both entropies are temporally constant as has been well known for decades. The addition of stochastic perturbations (a Wiener process) leads to an indeterminate (either increasing or decreasing) behaviour of the Gibbs' entropy, but the conditional entropy monotonically approaches equilibrium with increasing time. The presence of delays in the dynamics, whether stochastically perturbed or not, leads to situations in which the Gibbs' and conditional entropies evolution can be oscillatory and not monotone, and may not approach equilibrium.

math-ph↗
arXiv2018

The combined effects of Feller diffusion and transcriptional/translational bursting in simple gene networks

We study a stochastic model of biosynthesis of proteins in generic bacterial operons. The stochasticity arises from two different processes, namely from `bursting' production of either mRNA and/or protein (in the transcription/translation process) and from standard diffusive fluctuations. The amount of protein follows the Feller diffusion, while the bursting introduces random jumps between trajectories of the diffusion process. The combined effect leads to a process commonly known as a diffusion process with jumps. We study existence of invariant densities and the long time behavior of distributions of the corresponding Markov process, proving asymptotic stability in the evolution of the density.

q-bio.MN↗
arXiv2017

Self-similar solutions of fragmentation equations revisited

We study the large time behaviour of the mass (size) of particles described by the fragmentation equation with homogeneous breakup kernel. We give necessary and sufficient conditions for the convergence of solutions to the unique self-similar solution.

math.AP↗
arXiv2016

Response of an oscillatory differential delay equation to a single stimulus

Here we analytically examine the response of a limit cycle solution to a simple differential delay equation to a single pulse perturbation of the piecewise linear nonlinearity. We construct the unperturbed limit cycle analytically, and are able to completely characterize the perturbed response to a pulse of positive amplitude and duration with onset at different points in the limit cycle. We determine the perturbed minima and maxima and period of the limit cycle and show how the pulse modifies these from the unperturbed case.

math.DS↗
arXiv2015

Piecewise deterministic Markov processes in biological models

We present a short introduction into the framework of piecewise deterministic Markov processes. We illustrate the abstract mathematical setting with a series of examples related to dispersal of biological systems, cell cycle models, gene expression, physiologically structured populations, as well as neural activity. General results concerning asymptotic properties of stochastic semigroups induced by such Markov processes are applied to specific examples.

math.PR↗
arXiv2015

The Limiting Dynamics of a Bistable Molecular Switch With and Without Noise

We consider the dynamics of a population of organisms containing two mutually inhibitory gene regulatory networks, that can result in a bistable switch-like behaviour. We completely characterize their local and global dynamics in the absence of any noise, and then go on to consider the effects of either noise coming from bursting (transcription or translation), or Gaussian noise in molecular degradation rates when there is a dominant slow variable in the system. We show analytically how the steady state distribution in the population can range from a single unimodal distribution through a bimodal distribution and give the explicit analytic form for the invariant stationary density which is globally asymptotically stable. Rather remarkably, the behaviour of the stationary density with respect to the parameters characterizing the molecular behaviour of the bistable switch is qualitatively identical in the presence of noise coming from bursting as well as in the presence of Gaussian noise in the degradation rate. This implies that one cannot distinguish between either the dominant source or nature of noise based on the stationary molecular distribution in a population of cells. We finally show that the switch model with bursting but two dominant slow genes has an asymptotically stable stationary density.

q-bio.MN↗
arXiv2015

Existence of invariant densities for semiflows with jumps

The problem of existence and uniqueness of absolutely continuous invariant measures for a class of piecewise deterministic Markov processes is investigated using the theory of substochastic semigroups obtained through the Kato--Voigt perturbation theorem on the $L^1$-space. We provide a new criterion for the existence of a strictly positive and unique invariant density for such processes. The long time qualitative behavior of the corresponding semigroups is also considered. To illustrate our general results we give a detailed study of a two dimensional model of gene expression with bursting.

math.PR↗
arXiv2010

Convergence to Lévy stable processes under some weak dependence conditions

For a strictly stationary sequence of random vectors in $\mathbb{R}^d$ we study convergence of partial sum processes to Lévy stable process in the Skorohod space with $J_1$-topology. We identify necessary and sufficient conditions for such convergence and provide sufficient conditions when the stationary sequence is strongly mixing.

math.PR↗
arXiv2009

Substochastic semigroups and densities of piecewise deterministic Markov processes

Necessary and sufficient conditions are given for a substochastic semigroup on $L^1$ obtained through the Kato--Voigt perturbation theorem to be either stochastic or strongly stable. We show how such semigroups are related to piecewise deterministic Markov process, provide a probabilistic interpretation of our results, and apply them to fragmentation equations.

math.FA↗
arXiv2008

Dynamics and density evolution in piecewise deterministic growth processes

A new sufficient condition is proved for the existence of stochastic semigroups generated by the sum of two unbounded operators. It is applied to one-dimensional piecewise deterministic Markov processes, where we also discuss the existence of a unique stationary density and give sufficient conditions for asymptotic stability.

math.AP↗
arXiv2008

Central Limit Theorems for Non-Invertible Measure Preserving Maps

We establish a new functional central limit theorem result for non-invertible measure preserving maps that are not necessarily ergodic, using the Perron-Frobenius operator. We apply the result to asymptotically periodic transformations and give an extensive specific example of asymptotically periodic transformations by using the tent map.

math.PR↗
arXiv2005

Temporal Behavior of the Conditional and Gibbs' Entropies

We study the temporal approach to equilibrium of the Gibbs' and conditional entropies for both invertible deterministic dynamics as well as non-invertible stochastic systems in the presence of white noise. The conditional entropy will either remain constant or monotonically increase to its maximum of zero. However, the Gibbs' entropy may have a variety of patterns of approach to its final value ranging from a monotone increase or decrease to an oscillatory approach. We have illustrated all of these behaviors using examples in which both entropy dynamics can be determined analytically.

cond-mat.stat-mech↗
arXiv2005

Effects of Noise on Entropy Evolution

We study the convergence properties of the conditional (Kullback-Leibler) entropy in stochastic systems. We have proved very general results showing that asymptotic stability is a necessary and sufficient condition for the monotone convergence of the conditional entropy to its maximal value of zero. Additionally we have made specific calculations of the rate of convergence of this entropy to zero in a one-dimensional situations, illustrated by Ornstein-Uhlenbeck and Rayleigh processes, higher dimensional situations, and a two dimensional Ornstein-Uhlenbeck process with a stochastically perturbed harmonic oscillator and colored noise as examples. We also apply our general results to the problem of conditional entropy convergence in the presence of dichotomous noise. In both the single dimensional and multidimensional cases we are to show that the convergence of the conditional entropy to zero is monotone and at least exponential. In the specific cases of the Ornstein-Uhlenbeck and Rayleigh processes as well as the stochastically perturbed harmonic oscillator and colored noise examples, we have the rather surprising result that the rate of convergence of the entropy to zero is independent of the noise amplitude.

cond-mat.stat-mech↗
arXiv2004

An invariance principle for maps with polynomial decay of correlations

We give a general method of deriving statistical limit theorems, such as the central limit theorem and its functional version, in the setting of ergodic measure preserving transformations. This method is applicable in situations where the iterates of discrete time maps display a polynomial decay of correlations.

math.DS↗
arXiv2004

Deterministic Brownian Motion: The Effects of Perturbing a Dynamical System by a Chaotic Semi-Dynamical System

Here we review and extend central limit theorems for highly chaotic but deterministic semi-dynamical discrete time systems. We then apply these results show how Brownian motion-like results are recovered, and how an Ornstein-Uhlenbeck process results within a totally deterministic framework. These results illustrate that the contamination of experimental data by "noise" may, under certain circumstances, be alternately interpreted as the signature of an underlying chaotic process.

cond-mat.stat-mech↗