Related papers: Liouville property on $G$-spaces

Let $G$ be a compact, connected simple Lie group and $\mathfrak{g}$ its Lie algebra. It is known that if $\mu $ is any $G$-invariant measure supported on an adjoint orbit in $\mathfrak{g}$, then for each integer $k$, the $k$% -fold…

We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ defined by this action for some fixed generating set. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the…

For large classes of even-dimensional Riemannian manifolds $(M,g)$, we construct and analyze conformally invariant random fields. These centered Gaussian fields $h=h_g$, called co-polyharmonic Gaussian fields, are characterized by their…

We construct a Gibbs measure for the nonlinear Schrodinger equation (NLS) on the circle, conditioned on prescribed mass and momentum: d \mu_{a,b} = Z^{-1} 1_{\int_T |u|^2 = a} 1_{i \int_T u \bar{u}_x = b} exp (\pm1/p \int_T |u|^p - 1/2…

We introduce a strategy to tackle some known obstructions of current approaches to the Fourier uniformity conjecture. Assuming GRH, we then show the conjecture holds for intervals of length at least $(\log X)^{\psi(X)}$, with $\psi(X)…

We prove that if a Borel probability measure (\mu) on (\T) is invariant under the action of a "large" multiplicative semigroup (lower logarithmic density is positive) and the action of the whole semigroup is ergodic then (\mu) is either…

This is a survey article of geometric properties of noncommutative symmetric spaces of measurable operators $E(\mathcal{M},\tau)$, where $\mathcal{M}$ is a semifinite von Neumann algebra with a faithful, normal, semifinite trace $\tau$, and…

Standard statistical mechanics of conservative systems relies on the symplectic geometry of the phase space. This is exploited to derive Hamilton's equations, Liouville's theorem and to find the canonical invariant measure. In this work we…

The gravitational measure on an arbitrary topological three-manifold is constructed. The nontrivial dependence of the measure on the conformal factor is discussed. We show that only in the case of a compact manifold with boundary the…

We establish a Liouville-type theorem for the elliptic and incompressible Magnetic-B\'enard system defined over the entire three-dimensional space. Specifically, we demonstrate the uniqueness of trivial solutions under the condition that…

Let $(\pi, \mathcal H)$ be a continuous unitary representation of the (infinite dimensional) Lie group $G$ and $\gamma \: \mathbb R \to \mathrm{Aut}(G)$ define a continuous action of $\mathbb R$ on $G$. Suppose that $\pi^\#(g,t) = \pi(g)…

In this paper, we establish Liouville-type theorems for a one-parameter family of elliptic PDEs on the standard upper half-plane model of the hyperbolic space, under specific geometric assumptions. Our results indicate that the Euclidean…

Over the past few decades, two natural random surface models have emerged within physics and mathematics. The first is Liouville quantum gravity, which has its roots in string theory and conformal field theory from the 1980s and 1990s. The…

We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in Hult & Lindskog (2006), Das, Mitra & Resnick (2013). Our framework…

We show that the groups of finite energy loops and paths (that is, those of Sobolev class $H^1$) with values in a compact connected Lie group, as well as their central extensions, satisfy an amenability-like property: they admit a…

Let $G$ be a Lie group and $\Gamma$ be a lattice in $G$. We introduce the notion of locally unipotent invariant measures on $G/\Gamma$. We then prove that under some conditions, the limit measure supported on the image of polynomial…

The class of Sturm-Liouville operators on the space of square integrable functions on a finite interval is considered. According to the Riesz-spectral property, the self-adjointness and the positivity of such unbounded linear operators on…

In this work we study the Ruelle Operator associated to a continuous potential defined on a countable product of a compact metric space. We prove a generalization of Bowen's criterion for the uniqueness of the eigenmeasures. One of the main…

We prove many new cases of Zimmer's conjecture for actions by lattices in non-$\mathbb{R}$-split semisimple Lie groups $G$. By prior arguments, Zimmer's conjecture reduces to studying certain probability measures invariant under a minimal…

We consider the metric growth in Liouville quantum gravity (LQG) for $\gamma \in (0,2)$. We show that a process associated with the trace of the free field on the boundary of a filled LQG ball is stationary, for every $\gamma \in (0,2)$.…