Exponential ergodicity for regime-switching diffusion processes in total variation norm

<p style='text-indent:20px;'>We investigate the long time behavior for a class of regime-switching diffusion processes. Based on direct evaluation of moments and exponential functionals of hitting time of the underlying process, we adopt coupling method to obtain existence and uniqueness of the invariant probability measure and establish explicit exponential bounds for the rate of convergence to the invariant probability measure in total variation norm. In addition, we provide some concrete examples to illustrate our main results which reveal impact of random switching on stochastic stability and convergence rate of the system.</p>

Weighted L 2-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential U allowing for singularities. By modifying the direct approach to convergence in L 2 pioneered by Hérau and developed by Dolbeault et al, we show that the dynamics converges exponentially fast to equilibrium in the topologies L 2(dμ) and L 2(W* dμ), where μ denotes the invariant probability measure and W* is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter γ in Langevin dynamics, by providing a lower bound scaling as min(γ, γ −1). The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

Amenability, connected components, and definable actions

We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -definable group G (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of G implies (under some assumption) that $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

Sensitivity of steady states in a degenerately damped stochastic Lorenz system

We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz ’63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant probability measure if and only if noise acts on the convection variable. On the other hand, if there is a positive growth term on the vertical temperature profile, we prove that there is no normalizable invariant state. Our approach relies on the derivation and analysis of nontrivial Lyapunov functions which ensure positive recurrence or null-recurrence/transience of the dynamics.

Solution to the Quantum Symmetric Simple Exclusion Process: The Continuous Case

The quantum symmetric simple exclusion process (Q-SSEP) is a model for quantum stochastic dynamics of fermions hopping along the edges of a graph with Brownian noisy amplitudes and driven out-of-equilibrium by injection-extraction processes at a few vertices. We present a solution for the invariant probability measure of the one dimensional Q-SSEP in the infinite size limit by constructing the steady correlation functions of the system density matrix and quantum expectation values. These correlation functions code for a rich structure of fluctuating quantum correlations and coherences. Although our construction does not rely on the standard techniques from the theory of integrable systems, it is based on a remarkable interplay between the permutation groups and polynomials. We incidentally point out a possible combinatorial interpretation of the Q-SSEP correlation functions via a surprising connexion with geometric combinatorics and the associahedron polytopes.

Pfaffian Point Processes from Free Fermion Algebras: Perfectness and Conditional Measures

The analogy between determinantal point processes (DPPs) and free fermionic calculi is well-known. We point out that, from the perspective of free fermionic algebras, Pfaffian point processes (PfPPs) naturally emerge, and show that a positive contraction acting on a ''doubled'' one-particle space with an additional structure defines a unique PfPP. Recently, Olshanski inverted the direction from free fermions to DPPs, proposed a scheme to construct a fermionic state from a quasi-invariant probability measure, and introduced the notion of perfectness of a probability measure. We propose a method to check the perfectness and show that Schur measures are perfect as long as they are quasi-invariant under the action of the symmetric group. We also study conditional measures for PfPPs associated with projection operators. Consequently, we show that the conditional measures are again PfPPs associated with projection operators onto subspaces explicitly described.

On the fractional susceptibility function of piecewise expanding maps

<p style='text-indent:20px;'>We associate to a perturbation <inline-formula><tex-math id="M1">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> of a (stably mixing) piecewise expanding unimodal map <inline-formula><tex-math id="M2">\begin{document}$ f_0 $\end{document}</tex-math></inline-formula> a two-variable fractional susceptibility function <inline-formula><tex-math id="M3">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula>, depending also on a bounded observable <inline-formula><tex-math id="M4">\begin{document}$ \phi $\end{document}</tex-math></inline-formula>. For fixed <inline-formula><tex-math id="M5">\begin{document}$ \eta \in (0,1) $\end{document}</tex-math></inline-formula>, we show that the function <inline-formula><tex-math id="M6">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> is holomorphic in a disc <inline-formula><tex-math id="M7">\begin{document}$ D_\eta\subset \mathbb{C} $\end{document}</tex-math></inline-formula> centered at zero of radius <inline-formula><tex-math id="M8">\begin{document}$ &gt;1 $\end{document}</tex-math></inline-formula>, and that <inline-formula><tex-math id="M9">\begin{document}$ \Psi_\phi(\eta, 1) $\end{document}</tex-math></inline-formula> is the Marchaud fractional derivative of order <inline-formula><tex-math id="M10">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> of the function <inline-formula><tex-math id="M11">\begin{document}$ t\mapsto \mathcal{R}_\phi(t): = \int \phi(x)\, d\mu_t $\end{document}</tex-math></inline-formula>, at <inline-formula><tex-math id="M12">\begin{document}$ t = 0 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M13">\begin{document}$ \mu_t $\end{document}</tex-math></inline-formula> is the unique absolutely continuous invariant probability measure of <inline-formula><tex-math id="M14">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. In addition, we show that <inline-formula><tex-math id="M15">\begin{document}$ \Psi_\phi(\eta, z) $\end{document}</tex-math></inline-formula> admits a holomorphic extension to the domain <inline-formula><tex-math id="M16">\begin{document}$ \{\, (\eta, z) \in \mathbb{C}^2\mid 0&lt;\Re \eta &lt;1, \, z \in D_\eta \,\} $\end{document}</tex-math></inline-formula>. Finally, if the perturbation <inline-formula><tex-math id="M17">\begin{document}$ (f_t) $\end{document}</tex-math></inline-formula> is horizontal, we prove that <inline-formula><tex-math id="M18">\begin{document}$ \lim_{\eta \in (0,1), \eta \to 1}\Psi_\phi(\eta, 1) = \partial_t \mathcal{R}_\phi(t)|_{t = 0} $\end{document}</tex-math></inline-formula>.</p>

Equitable colourings of Borel graphs

Hajnal and Szemerédi proved that if G is a finite graph with maximum degree $\Delta $ , then for every integer $k \geq \Delta +1$ , G has a proper colouring with k colours in which every two colour classes differ in size at most by $1$ ; such colourings are called equitable. We obtain an analogue of this result for infinite graphs in the Borel setting. Specifically, we show that if G is an aperiodic Borel graph of finite maximum degree $\Delta $ , then for each $k \geq \Delta + 1$ , G has a Borel proper k-colouring in which every two colour classes are related by an element of the Borel full semigroup of G. In particular, such colourings are equitable with respect to every G-invariant probability measure. We also establish a measurable version of a result of Kostochka and Nakprasit on equitable $\Delta $ -colourings of graphs with small average degree. Namely, we prove that if $\Delta \geq 3$ , G does not contain a clique on $\Delta + 1$ vertices and $\mu $ is an atomless G-invariant probability measure such that the average degree of G with respect to $\mu $ is at most $\Delta /5$ , then G has a $\mu $ -equitable $\Delta $ -colouring. As steps toward the proof of this result, we establish measurable and list-colouring extensions of a strengthening of Brooks’ theorem due to Kostochka and Nakprasit.

Positive Entropy Using Hecke Operators at a Single Place

We prove the following statement: let $X=\textrm{SL}_n({{\mathbb{Z}}})\backslash \textrm{SL}_n({{\mathbb{R}}})$ and consider the standard action of the diagonal group $A&lt;\textrm{SL}_n({{\mathbb{R}}})$ on it. Let $\mu $ be an $A$-invariant probability measure on $X$, which is a limit $$\begin{equation*} \mu=\lambda\lim_i|\phi_i|^2dx, \end{equation*}$$where $\phi _i$ are normalized eigenfunctions of the Hecke algebra at some fixed place $p$ and $\lambda&gt;0$ is some positive constant. Then any regular element $a\in A$ acts on $\mu $ with positive entropy on almost every ergodic component. We also prove a similar result for lattices coming from division algebras over ${{\mathbb{Q}}}$ and derive a quantum unique ergodicity result for the associated locally symmetric spaces. This generalizes a result of Brooks and Lindenstrauss [2].