Stochastic homogenization and ergodic theory

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

Download Full-text

Ergodic theory for energetically open compressible fluid flows

Physica D Nonlinear Phenomena ◽

10.1016/j.physd.2021.132914 ◽

2021 ◽

Vol 423 ◽

pp. 132914

Author(s):

Francesco Fanelli ◽

Eduard Feireisl ◽

Martina Hofmanová

Keyword(s):

Ergodic Theory ◽

Compressible Fluid ◽

Fluid Flows ◽

Compressible Fluid Flows

Download Full-text

Category Theory for Autonomous and Networked Dynamical Systems

Entropy ◽

10.3390/e21030302 ◽

2019 ◽

Vol 21 (3) ◽

pp. 302 ◽

Cited By ~ 4

Author(s):

Jean-Charles Delvenne

Keyword(s):

Dynamical Systems ◽

Systems Theory ◽

Ergodic Theory ◽

Category Theory ◽

Open Systems ◽

Topological Dynamics ◽

Control Problems ◽

Natural Conditions ◽

Discussion Paper ◽

Open Systems Theory

In this discussion paper we argue that category theory may play a useful role in formulating, and perhaps proving, results in ergodic theory, topogical dynamics and open systems theory (control theory). As examples, we show how to characterize Kolmogorov–Sinai, Shannon entropy and topological entropy as the unique functors to the nonnegative reals satisfying some natural conditions. We also provide a purely categorical proof of the existence of the maximal equicontinuous factor in topological dynamics. We then show how to define open systems (that can interact with their environment), interconnect them, and define control problems for them in a unified way.

Download Full-text