相关论文: Time functions and $K$-causality between measures

Let M and \bar M be n-dimensional manifolds equipped with suitable Borel probability measures \rho and \bar\rho. Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c \in C^4(M \times \bar M) to guarantee smoothness of…

In this article we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measuresof Iterated Function Systems equipped with a probability distribution. We recover a classical…

When studying the causal propagation of a field in a globally hyperbolic spacetime M, one often wants to express the physical intuition that it has compact support in spacelike directions, or that its support is a spacelike compact set. We…

We establish connections between the requirement of measurability of a probability space and the principle of complimentarity in quantum mechanics. It is shown that measurability of a probability space implies the dependence of results of…

Topology change is considered to be a necessary feature of quantum gravity by some authors, and impossible by others. One of the main arguments against it is that spacetimes with changing spatial topology have bad causal properties. Borde…

Planck scale modified dispersion relations are one way how to capture the influence of quantum gravity on the propagation of fundamental point particles effectively. We derive the time dilation between an observer's or particle's proper…

The mutual information characterizes correlations between spatially separated regions of a system. Yet, in experiments we often measure dynamical correlations, which involve probing operators that are also separated in time. Here, we…

Determining and measuring cause-effect relationships is fundamental to most scientific studies of natural phenomena. The notion of causation is distinctly different from correlation which only looks at association of trends or patterns in…

Measures of the direction and strength of the interdependence between two time series are evaluated and modified in order to reduce the bias in the estimation of the measures, so that they give zero values when there is no causal effect.…

We extend the classical Mercer theorem to reproducing kernel Hilbert spaces whose elements are functions from a measurable space $X$into $\mathbb C^n$. Given a finite measure $\mu$ on $X$, we represent the reproducing kernel $K$ as…

The relativistic conception of space and time is challenged by the quantum nature of physical observables. It has been known for a long time that Poincar\'e symmetry of field theory can be extended to the larger conformal symmetry. We use…

We consider the simultaneous optimal transportation of measures, where the target marginal is not necessarily fixed. For this problem, we prove the existence of a solution for completely regular spaces and investigate the structure of the…

Several other factors, besides the intrinsic local geometry, contribute to give a meaning to a space-time model. The simplest example comes from comparing Minkowski's and Milne's model, that both have a null Riemann tensor. We add to these…

The main aim of the paper is to introduce a new class of (semigroup-valued) measures that are ultrahomogeneous on the Boolean algebra of all clopen subsets of the Cantor space and to study their automorphism groups. A characterisation, in…

Studies in 1+1 dimensions suggest that causally discontinuous topology changing spacetimes are suppressed in quantum gravity. Borde and Sorkin have conjectured that causal discontinuities are associated precisely with index 1 or n-1 Morse…

Quantum mechanics introduces the concept of probability at the fundamental level, yielding the measurement problem. On the other hand, recent progress in cosmology has led to the "multiverse" picture, in which our observed universe is only…

Given two time series, can one tell, in a rigorous and quantitative way, the cause and effect between them? Based on a recently rigorized physical notion namely information flow, we arrive at a concise formula and give this challenging…

Causality is one of the fundamental structures of spacetimes, determining the possible behaviour and propagation of physical information. Causal structure can be analysed through the various modal logics it induces. The modal logics for the…

This paper proposes a Multimarginal Optimal Transport ($MOT$) approach for simultaneously comparing $k\geq 2$ measures supported on finite subsets of $\mathbb{R}^d$, $d \geq 1$. We derive asymptotic distributions of the optimal value of the…

We show how the measure theory of regular compacted-Borel measures defined on the $\delta$-ring of compacted-Borel subsets of a weighted locally compact group $(G,\omega)$ provides a compatible framework for defining the corresponding…