Transportation Inequalities Research Articles

Travel satisfaction inequality and the role of the urban metro system

Transport equality is one of the key aspects of sustainable urban transport. Personal satisfaction with travel among different communities and social groups has become a main theme in the field of transport equality research. However, there has been little investigation of the expectation confirmation mechanism underneath this affective feeling. Moreover, there is not enough evidence on how the discrepancy between expected and actual travel relates to satisfaction. This study first identifies the regional and social disparity in general travel satisfaction among residents in Beijing, then discusses how the satisfaction response to the dissonance between expected and actual travel, and finally investigates the role of urban rail transport in this disparity. The results show that general travel satisfaction presents both regional and social disparities, with city-centre and middle-income residents having higher travel satisfaction. Disconfirmation of travel expectations partly explains travel dissatisfaction. The inconsistency between travel discrepancy level and satisfaction for lower income residents further reflects transport inequality issues, and it provides a clue for transport policymakers to promote transport equity from a sufficientarianism perspective. Public transport-related development has an important role in improving different aspects of travel satisfaction. Among them, higher level of mixed land use in metro station catchment areas and higher metro network centrality of the home nearest station help to increase the travel satisfaction of disadvantaged residents. This is realised via enhancing destination options for low-income or suburban residents, and via providing a relatively smooth and tidy travel experience for disadvantaged groups, who tend to use metro as a major travel mode.

Examining the Equity Impacts of Autonomous Vehicles: A Travel Demand Model Approach

As investments in autonomous vehicle (AV) technology continue to grow, agencies are beginning to consider how AVs will affect travel behavior within their jurisdictions and how to respond to this new mobility technology. Different autonomous futures could reduce, perpetuate, or exacerbate existing transportation inequities. This paper presents a regional travel demand model used to quantify how transportation outcomes may differ for disadvantaged populations in the Washington, D.C. area under a variety of future scenarios. Transportation performance measures examined included job accessibility, trip duration, trip distance, mode share, and vehicle miles traveled. The model evaluated changes in these indicators for disadvantaged and non-disadvantaged communities under scenarios when AVs were primarily single-occupancy or high-occupancy, and according to whether transit agencies responded to AVs by maintaining the status quo, removing low-performing routes, or applying AV technology to transit vehicles. Across the performance measures, the high-occupancy AV and enhanced transit scenarios provided an equity benefit, either mitigating an existing gap in outcomes between demographic groups or reducing the extent to which that gap was expanded.

On the convex Poincaré inequality and weak transportation inequalities

We prove that for a probability measure on $\mathbb{R}^n$, the Poincar\'e inequality for convex functions is equivalent to the weak transportation inequality with a quadratic-linear cost. This generalizes recent results by Gozlan et al. and Feldheim et al., concerning probability measures on the real line. The proof relies on modified logarithmic Sobolev inequalities of Bobkov-Ledoux type for convex and concave functions, which are of independent interest. We also present refined concentration inequalities for general (not necessarily Lipschitz) convex functions, complementing recent results by Bobkov, Nayar and Tetali.

On the eigenvalues of truncations of random unitary matrices

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and Réffy identified the limiting spectral measure if $\frac{m} {n}\to \alpha $, as $n\to \infty $; under suitable scaling, the family $\{\mu _{\alpha }\}_{\alpha \in (0,1)}$ of limiting measures interpolates between uniform measure on the unit disc (for small $\alpha $) and uniform measure on the unit circle (as $\alpha \to 1$). In this note, we prove an explicit concentration inequality which shows that for fixed $n$ and $m$, the bounded-Lipschitz distance between the empirical spectral measure and the corresponding $\mu _{\alpha }$ is typically of order $\sqrt{\frac {\log (m)}{m}} $ or smaller. The approach is via the theory of two-dimensional Coulomb gases and makes use of a new “Coulomb transport inequality” due to Chafaï, Hardy, and Maïda.

Spherical Hellinger--Kantorovich Gradient Flows

We study nonlinear degenerate parabolic equations of Fokker--Planck type that can be viewed as gradient flows with respect to the recently introduced spherical Hellinger--Kantorovich distance. The driving entropy is not assumed to be geodesically convex. We prove solvability of the problem and the entropy-entropy production inequality, which implies exponential convergence to the equilibrium. As a corollary, we obtain some related results for the Wasserstein gradient flows. We also deduce transportation inequalities in the spirit of Talagrand, Otto, and Villani for the spherical and conic Hellinger--Kantorovich distances.

A second order equation for Schrödinger bridges with applications to the hot gas experiment and entropic transportation cost

The Schrodinger problem is obtained by replacing the mean square distance with the relative entropy in the Monge–Kantorovich problem. It was first addressed by Schrodinger as the problem of describing the most likely evolution of a large number of Brownian particles conditioned to reach an “unexpected configuration”. Its optimal value, the entropic transportation cost, and its optimal solution, the Schrodinger bridge, stand as the natural probabilistic counterparts to the transportation cost and displacement interpolation. Moreover, they provide a natural way of lifting from the point to the measure setting the concept of Brownian bridge. In this article, we prove that the Schrodinger bridge solves a second order equation in the Riemannian structure of optimal transport. Roughly speaking, the equation says that its acceleration is the gradient of the Fisher information. Using this result, we obtain a fine quantitative description of the dynamics, and a new functional inequality for the entropic transportation cost, that generalize Talagrand’s transportation inequality. Finally, we study the convexity of the Fisher information along Schrodigner bridges, under the hypothesis that the associated reciprocal characteristic is convex. The techniques developed in this article are also well suited to study the Feynman–Kac penalisations of Brownian motion.

Revisiting inequalities in the commuting burden: Institutional constraints and job-housing relationships in Beijing

Promoting social equality in transport has become a key concern in building inclusive cities. Inequities in the burden of commuting have attracted increasing attention from researchers. Although many studies have described the facts of transport inequality, there has been little in-depth research on institutional factors that influence commuting inequality. This paper aims to contribute fresh evidence on this topic by examining Beijing as a case study through a combination of quantitative and qualitative approaches. The results of this analysis show that increasing inequality in the commuting burden largely results from existing institutional discrimination against migrants, especially highly-skilled migrants. Beijing's Danwei, Hukou and BGC systems, as well as the city's housing distribution system, create serious institutional barriers to maintaining Beijing's jobs-housing balance, causing extra commuting burdens for highly-skilled migrants. Surprisingly, low-skilled migrants are restricted less by institutional barriers and maintain a good job-housing relationship due to sufficient provisions of informal houses and jobs in a broad range of urban areas. Our study reveals a “back to Danwei house” movement through which local households negotiate the growing commuting burden with the pursuit of quality of life. Based on these findings, we argue that increasing social inequality in the commuting burden may continue to worsen unless institutional discrimination against disadvantaged groups is removed in the process of China's economic transition.

Concentration for Coulomb gases and Coulomb transport inequalities

We study the non-asymptotic behavior of Coulomb gases in dimension two and more. Such gases are modeled by an exchangeable Boltzmann–Gibbs measure with a singular two-body interaction. We obtain concentration of measure inequalities for the empirical distribution of such gases around their equilibrium measure, with respect to bounded Lipschitz and Wasserstein distances. This implies macroscopic as well as mesoscopic convergence in such distances. In particular, we improve the concentration inequalities known for the empirical spectral distribution of Ginibre random matrices. Our approach is remarkably simple and bypasses the use of renormalized energy. It crucially relies on new inequalities between probability metrics, including Coulomb transport inequalities which can be of independent interest. Our work is inspired by the one of Maïda and Maurel-Segala, itself inspired by large deviations techniques. Our approach allows to recover, extend, and simplify previous results by Rougerie and Serfaty.

Transportation inequalities for stochastic heat equations

Using the method of Girsanov transformation, we establish the transportation inequalities, on the continuous paths space with respect to the L2 metric, for the law of the solution of stochastic heat equation driven by space–time white noise. The case of stochastic heat equations driven by fractional noise is also investigated.

Liquidity, Risk Measures, and Concentration of Measure

This paper studies curves of the form (ρ(λX))λ≥0, called risk profiles, where ρ is a convex risk measure and X a random variable. Financially, this captures the sensitivity of risk to the size of the investment in X, which the original axiomatic foundations of convex risk measures suggest to interpret as liquidity risk. The shape of a risk profile is intimately linked with the tail behavior of X for some notable classes of risk measures, namely shortfall risk measures and optimized certainty equivalents, and shares many useful features with cumulant generating functions. Exploiting this link leads to tractable necessary and sufficient conditions for pointwise bounds on risk profiles, which we call concentration inequalities. These inequalities admit useful dual representations related to transport inequalities, and this leads to efficient uniform bounds for risk profiles for large classes of X. Several interesting mathematical results emerge from this analysis, including a new perspective on nonexponential concentration estimates and some peculiar characterizations of classical transport inequalities. Lastly, the analysis is deepened by means of a surprising connection between time consistency properties of law invariant risk measures and the tensorization of concentration inequalities.

Transportation inequalities for fractional stochastic functional differential equations driven by fractional Brownian motion

Using the method of Girsanov transformation, we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution of fractional stochastic functional differential equations driven by fractional Brownian motion. The case of neutral fractional stochastic functional differential equations are also investigated.

Curvature and transport inequalities for Markov chains in discrete spaces

We study various transport-information inequalities under three different notions of Ricci curvature in the discrete setting: the curvature-dimension condition of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer), the exponential curvature-dimension condition of Bauer et al. (Li-Yau Inequality on Graphs (2013)) and the coarse Ricci curvature of Ollivier (J. Funct. Anal. 256 (2009) 810–864). We prove that under a curvature-dimension condition or coarse Ricci curvature condition, an $L_{1}$ transport-information inequality holds; while under an exponential curvature-dimension condition, some weak-transport information inequalities hold. As an application, we establish a Bonnet–Myers theorem under the curvature-dimension condition $\operatorname{CD}(\kappa,\infty)$ of Bakry and Emery (In Seminaire de Probabilites, XIX, 1983/84 (1985) 177–206 Springer).

A variational approach to some transport inequalities

Dans cet article, nous proposons une approche des inegalites de transport-entropie fondee sur la minimisation de certaines fonctionnelles definies sur l’espace des mesures de probabilite. Cette approche nous permet en particulier de donner une nouvelle preuve d’un resultat d’Otto et Villani (J. Funct. Anal. 173 (2000) 361–400) montrant que l’inegalite de Sobolev logarithmique entraine l’inegalite de transport de Talagrand.

Transportation and concentration inequalities for bifurcating Markov chains

We investigate the transportation inequality for bifurcating Markov chains which are a class of processes indexed by a regular binary tree. Fitting well models like cell growth when each individual gives birth to exactly two offsprings, we use transportation inequalities to provide useful concentration inequalities. We also study deviation inequalities for the empirical means under relaxed assumptions on the Wasserstein contraction for the Markov kernels. Applications to bifurcating nonlinear autoregressive processes are considered for point-wise estimates of the non-linear autoregressive function.

Transportation Inequalities for Neutral Stochastic Differential Equations Driven by Fractional Brownian Motion with Hurst Parameter Lesser Than 1/2

In this note, we prove an existence and uniqueness result of mild solution for a neutral stochastic differential equation with finite delay, driven by a fractional Brownian motion with Hurst parameter lesser than 1/2 in a Hilbert space, and we establish the transportation inequalities, with respect to the uniform distance, for the law of the mild solution.

Transportation Inequalities for Multivalued Stochastic Evolution Equations

Transportation Inequalities for Multivalued Stochastic Evolution Equations

Transport Inequalities for Log-concave Measures, Quantitative Forms, and Applications

AbstractWe review some simple techniques based on monotonemass transport that allow us to obtain transport-type inequalities for any log-concave probability measures, and formore generalmeasures as well. We discuss quantitative forms of these inequalities, with application to the Brascamp-Lieb variance inequality.

Hamilton-Jacobi Equations on Graph and Applications

This paper introduces a notion of gradient and an infimal-convolution operator that extend properties of solutions of Hamilton Jacobi equations to more general spaces, in particular to graphs. As a main application, the hypercontractivity of this class of infimal-convolution operators is connected to some discrete version of the log-Sobolev inequality and to a discrete version of Talagrand’s transport inequality.

On the sub-Gaussianity of the Beta and Dirichlet distributions

We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.

Hitting times, functional inequalities, Lyapunov conditions and uniform ergodicity

The use of Lyapunov conditions for proving functional inequalities was initiated in [5]. It was shown in [4,30] that there is an equivalence between a Poincaré inequality, the existence of some Lyapunov function and the exponential integrability of hitting times. In the present paper, we close the scheme of the interplay between Lyapunov conditions and functional inequalities by•showing that strong functional inequalities are equivalent to Lyapunov type conditions;•showing that these Lyapunov conditions are characterized by the finiteness of generalized exponential moments of hitting times. We also give some complement concerning the link between Lyapunov conditions and integrability property of the invariant probability measure and as such transportation inequalities, and we show that some “unbounded Lyapunov conditions” can lead to uniform ergodicity, and coming down from infinity property.