We establish a one-to-one correspondence between one-sided and two-sided regular systems of conditional probabilities on the half-line that preserves the associated chains and Gibbs measures. As an application, we determine uniqueness and non-uniqueness regimes in one-sided versions of ferromagnetic Ising models with long range interactions. Our study shows that the interplay between chain and Gibbsian theories yields more information than that contained within the known theory of each separate framework. In particular: (i) A Gibbsian construction due to Dyson yields a new family of chains with phase transitions; (ii) these transitions show that a square summability uniqueness condition of chains is false in the general non-shift-invariant setting, and (iii) an uniqueness criterion for chains shows that a Gibbsian conjecture due to Kac and Thompson is false in this half-line setting.
Non-uniqueness of three-dimensional Euler equations and Navier-Stokes equations forced by random noise, path-wise and more recently even in law, have been proven by various authors. We prove non-uniqueness in law of the three-dimensional Navier-Stokes equations forced by random noise and diffused via a fractional Laplacian that has power between zero and one half. The solution we construct has H$\ddot{\mathrm{o}}$lder regularity with a small exponent rather than Sobolev regularity with a small exponent. For the power sufficiently small, the non-uniqueness in law holds at the level of Leray-Hop…
In the present paper, we consider a class of quadratic stochastic operators (q.s.o.) called $ b- $bistochastic q.s.o. We include several properties of $ b- $bistochastic q.s.o. and their dynamical behavior. One of the main findings in this paper is the description on the uniqueness of the fixed points. Besides, we list the conditions on strict contractive $ b- $bistochastic q.s.o. on low dimensional simplices and it turns out that, the uniqueness of the fixed point does not imply strict contraction. Finally, we associated Markov measures with $b$-bistochastic q.s.o. On a class of $b$-bistochas…
In this paper we introduce a notion of asymptotic stability of a probability kernel, which we call dynamic uniqueness. We say that a kernel exhibits dynamic uniqueness if all the stochastic chains starting from a fixed past coincide on the future tail $σ$-algebra. We prove that the dynamic uniqueness is generally stronger than the usual notion of uniqueness for $g$-measures. Our main result shows that dynamic uniqueness is equivalent to the weak-$\ell^2$ summability condition on the kernel. This generalizes and strengthens the Johansson-Öberg $\ell^2$ criterion for uniqueness of $g$-measures. …
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising model on a graph $G=(V,E)$. The dynamics is a "global" Markov chain which is conjectured to converge to equilibrium in $O(|V|^{1/4})$ steps for any graph $G$ at any (inverse) temperature $β$. It was recently proved by Guo and Jerrum (2017) that the Swendsen-Wang dynamics has polynomial mixing time on any graph at all temperatures, yet there are few results providing $o(|V|)$ upper bounds on its convergence time. We prove fast convergence of the Swendsen-Wang dynamics on genera…
In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive-attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the $λ-$Wasserstein metric topology with $1\le λ<\infty$, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of $\infty$-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.
