by JEAN-PIERRE ECKMANN (Geneva) and JEREMY QUASTEL (Toronto) Before starting to work on KPZ and regularity structures, Martin’s papers covered a wide range of important fields: Stochastic PDE’s in infinite dimensions [3, 4, 7], uniqueness of the invariant measure for 2D Navier Stokes [5], transport properties of very singular heat conduction problems [6]. And it is perhaps of interest to point ...

An essential spanning forest of an infinite graph G is a spanning forest of G in which all trees have infinitely many vertices. Let Gn be an increasing sequence of finite connected subgraphs of G for which ∪Gn = G. Pemantle’s arguments (1991) imply that the uniform measures on spanning trees of Gn converge weakly to an Aut(G)-invariant measure μG on essential spanning forests of G. We show that...

A classical result due to Eagleson states (in particular) that if appropriately normalized Birkhoff sums generated by a measurable function and an ergodic probability preserving transformation converge in distribution, then they also distribution with respect any measure which is absolutely continuous the invariant one. In this note, we prove several quantitative infinite-dimensional versions o...

In this paper we investigate the problem of the existence of invariant measures on the local gauge group. We prove that it is impossible to define a finite translationally invariant measure on the local gauge group C∞(Rn, G)(where G is an arbitrary matrix Lie group). Functional integral over the local gauge group is a commonly employed technique in the study of gauge field theories. Its earlies...

We define the notion of uniformly recurrent subgroup, URS in short, which is a topological analog of the notion of invariant random subgroup (IRS), introduced in [2]. Our main results are as follows. (i) It was shown in [26] that for an arbitrary countable infinite group G, any free ergodic probability measure preserving G-system admits a minimal model. In contrast we show here, using URS’s, th...

We introduce the notion of aperiodicity measure for in nite symbolic sequences. Informally speaking, the aperiodicity measure of a sequence is the maximum number (between 0 and 1) such that this sequence di ers from each of its non-identical shifts in at least fraction of symbols being this number. We give lower and upper bounds on the aperiodicity measure of a sequence over a xed alphabet. We ...

magnetotelluric (mt) method is an electromagnetic technique that uses the earth natural field to map the electrical resistivity changes in subsurface structures. because of the high penetration depth of the electromagnetic fields in this method (tens of meters to tens of kilometers), the mt data is used to investigate the shallow to deep subsurface geoelectrical structures and their dimensions....

let be a locally compact non?abelian group and be a compact subgroup of also let be a ?invariant measure on the homogeneous space . in this article, we extend the linear operator as a bounded surjective linear operator for all ?spaces with . as an application of this extension, we show that each frame for determines a frame for and each frame for arises from a frame in via the linear operator .

For a continuous action of a countable discrete group G on a Polish space X, a countable Borel partition P of X is called a generator if GP ∶= {gP ∶ g ∈ G,P ∈ P} generates the Borel σ-algebra of X. For G = Z, the Kolmogorov–Sinai theorem gives a measuretheoretic obstruction to the existence of finite generators: they do not exist in the presence of an invariant probability measure with infinite...

In this article, we use two concepts, measure of non-compactness and Meir-Keeler condensing operators. The measure of non-compactness has been applied for existence of solution nonlinear integral equations, ordinary differential equations and system of differential equations in the case of finite and infinite dimensions by some authors. Also Meir-Keeler condensing operators are shown in some pa...