Space Of Finite Borel Measures Research Articles

The homology core of matchbox manifolds and invariant measures

We consider the topology and dynamics associated with a wide class of matchbox manifolds, including spaces of aperiodic tilings and suspensions of higher rank (potentially nonabelian) group actions on zero-dimensional spaces. For such a space we introduce a topological invariant, the homology core, built using an expansion of it as an inverse sequence of simplicial complexes. The invariant takes the form of a monoid equipped with a representation, which in many cases can be used to obtain a finer classification than is possible with the previously developed invariants. When the space is obtained by suspending a topologically transitive action of the fundamental group $ \Gamma $ of a closed orientable manifold on a zero-dimensional compact space $ Z$, this invariant corresponds to the space of finite Borel measures on $ Z$ which are invariant under the action of $ \Gamma $. This leads to connections between the rank of the core and the number of invariant, ergodic Borel probability measures for such actions. We illustrate with several examples how these invariants can be calculated and used for topological classification and how it leads to an understanding of the invariant measures.

Existence, Uniqueness and Lipschitz Dependence for Patlak–Keller–Segel and Navier–Stokes in $${\mathbb{R}^2}$$ R 2 with Measure-Valued Initial Data

We establish a new local well-posedness result in the space of finite Borel measures for mild solutions of the parabolic–elliptic Patlak–Keller–Segel (PKS) model of chemotactic aggregation in two dimensions. Our result only requires that the initial measure satisfy the necessary assumption $${\max_{x \in \mathbb{R}^2} \mu (\{x\}) < 8 \pi}$$ . This work improves the small-data results of Biler (Stud Math 114(2):181–192, 1995) and the existence results of Senba and Suzuki (J Funct Anal 191:17–51, 2002). Our work is based on that of Gallagher and Gallay (Math Ann 332:287–327, 2005), who prove the uniqueness and log-Lipschitz continuity of the solution map for the 2D Navier–Stokes equations (NSE) with measure-valued initial vorticity. We refine their techniques and present an alternative version of their proof which yields existence, uniqueness and Lipschitz continuity of the solution maps of both PKS and NSE. Many steps are more difficult for PKS than for NSE, particularly on the level of the linear estimates related to the self-similar spreading solutions.

Perron-Frobenius operators and the Klein-Gordon equation

For a smooth curve \Gamma and a set \Lambda in the plane \mathbb R^2 , let \mathrm{AC}(\Gamma;\Lambda) be the space of finite Borel measures in the plane supported on \Gamma , absolutely continuous with respect to the arc length and whose Fourier transform vanishes on \Lambda . Following [12], we say that (\Gamma,\Lambda) is a Heisenberg uniqueness pair if \mathrm{AC}(\Gamma;\Lambda)=\{0\} . In the context of a hyperbola \Gamma , the study of Heisenberg uniqueness pairs is the same as looking for uniqueness sets \Lambda of a collection of solutions to the Klein–Gordon equation. In this work, we mainly address the issue of finding the dimension of \mathrm{AC}(\Gamma;\Lambda) when it is non-zero. We will fix the curve \Gamma to be the hyperbola x_1x_2=1 , and the set \Lambda= \Lambda_{\alpha,\beta} to be the lattice-cross \Lambda_{\alpha,\beta}=\left(\alpha \mathbb Z\times\{0\}\right)\cup \left(\{0\}\times\beta \mathbb Z\right), where \alpha,\beta are positive reals. We will also consider \Gamma_+ , the branch of x_1x_2=1 where x_1&gt;0 . In [12], it is shown that \mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta})=\{0\} if and only if \alpha\beta\le1 . Here, we show that for \alpha\beta&gt;1 , we get a rather drastic “phase transition”: \mathrm{AC}(\Gamma;\Lambda_{\alpha,\beta}) is infinite-dimensional whenever \alpha\beta&gt;1 . It is shown in [13] that \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta})=\{0\} if and only if \alpha\beta&lt;4 . Moreover, at the edge \alpha\beta=4 , the behavior is more exotic: the space \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is one-dimensional. Here, we show that the dimension of \mathrm{AC}(\Gamma_+;\Lambda_{\alpha,\beta}) is infinite whenever \alpha\beta&gt;4 . Dynamical systems, and more specifically Perron–Frobenius operators, will play a prominent role in the presentation.

An Ergodic Decomposition Defined by Regular Jointly Measurable Markov Semigroups on Polish Spaces

For a regular jointly measurable Markov semigroup on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrisation of the ergodic probability measures associated to this semigroup in terms of subsets of the state space. In this way we extend results by Costa and Dufour (J. Appl. Probab. 43:767---781, 2006). As a consequence we obtain an integral decomposition of every invariant probability measure in terms of the ergodic probability measures. Our approach is completely centered around the reduction to and relationship with the case of a single regular Markov operator associated to the Markov semigroup, the resolvent operator, which enables us to fully exploit results in that situation (Worm and Hille in Ergod. Theory Dyn. Syst. 31(2):571---597, 2011).

Ergodic decompositions associated with regular Markov operators on Polish spaces

AbstractFor any regular Markov operator on the space of finite Borel measures on a Polish space we give a Yosida-type decomposition of the state space, which yields a parametrization of the ergodic probability measures associated with this operator in terms of particular subsets of the state space. We use this parametrization to prove an integral decomposition of every invariant probability measure in terms of the ergodic probability measures and give an ergodic decomposition of the state space. This extends results by Yosida [Functional Analysis. Springer, Berlin, 1980, Ch. XIII.4], Hernández-Lerma and Lasserre [Ergodic theorems and ergodic decomposition for Markov chains. Acta Appl. Math.54 (1998), 99–119] and Zaharopol [An ergodic decomposition defined by transition probabilities. Acta Appl. Math.104 (2008), 47–81], who considered the setting of locally compact separable metric spaces. Our extension to Polish spaces solves an open problem posed by Zaharopol (loc. cit.) in a satisfactory manner.

Periodic stochastic Korteweg–de Vries equation with additive space-time white noise

We prove the local well-posedness of the periodic stochastic Korteweg-de Vries equation with the additive space-time white noise. In order to treat low regularity of the white noise in space, we consider the Cauchy problem in the Besov-type space bb s∞(T) for s = 1 +, p = 2+ such that sp < 1. In establishing the local well-posedness, we use a variant of the Bourgain space adapted to b s∞(T) and establish a nonlinear estimate on the second iteration on the integral formulation. The deterministic part of the nonlinear estimate also yields the local well-posedness of the deterministic KdV in M(T), the space of finite Borel measures on T.