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Xiaojun Cui

Publications and source records attributed to Xiaojun Cui.

At least 19 recordsLinked to original sources
arXiv2026

Preimage Regions of Symmetric Separable Maps on the Simplex: Convexity and Barycentric Star-Shapedness

We study preimage regions on the open probability simplex associated with symmetric separable functionally generated maps. The problem is a finite-dimensional geometric question about convexity and barycentric star-shapedness of these regions. In the portfolio interpretation, the regions consist of the points whose generated portfolio has no negative coordinate. For symmetric separable generators, the defining first-order inequalities split into a coordinate term and a symmetric aggregation term. This coordinate--aggregation decomposition is the main organizing device of the paper. We show that the aggregation term may destroy convexity, and may even destroy barycentric star-shapedness. In particular, moving closer to the barycenter need not preserve the long-only property. We then give a necessary and sufficient threshold criterion for barycentric star-shapedness and derive sufficient conditions that recover it. These conditions are expressed in terms of concavity and second-derivative domination for the aggregation function. The entropy case is the affine aggregation case, in which the long-only constraints reduce to coordinate thresholds.

math.MG↗
arXiv2026

The Geometry of Admissible Short Selling in Discrete-Time Stochastic Portfolio Theory

While discrete-time Stochastic Portfolio Theory (SPT) provides a robust framework for market analysis, existing work on functional generation has predominantly focused on long-only portfolios defined on the entire unit simplex. This paper extends the geometric framework of functional generation to the broader class of bankruptcy-proof long-short portfolios defined on local market state spaces. We establish that, within this admissible setting, pseudo-arbitrage is fully characterized by the concavity of the generating function on the market state space, thereby relaxing the usual global domain requirement. A central contribution of this work is a geometric characterization of the short-selling mechanism. We prove that the presence of short selling is equivalent to the negativity of the maximal concave extension of the generating potential. This phenomenon is linked to the steepness of the logarithmic gradient as the market approaches a zero boundary nested inside the simplex. To systematically exploit this mechanism, we introduce the barycentric scaling transformation, a constructive methodology that maps classical long-only generating functions onto restricted domains to engineer admissible strategies with controlled short-selling exposure. Finally, through the analysis of specific shrunken portfolios, we identify a geometric phase transition: under suitable boundary conditions, admissible strategies exhibit a long-only core and a short-selling region in a qualitative sense (without asserting an exact partition of the state space). This provides a unified geometric perspective on relative arbitrage beyond the long-only constraint.

math.OC↗
arXiv2024

Global viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times

In this paper, we show that any globally hyperbolic space-time admits at least one globally defined distance-like function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave, as the one in the Riemannian case. Also, if the time orientation of a viscosity solution is non-consistent, we analyse its peculiar properties which make this kind of viscosity solutions are totally different from the ones where the Hamiltonians are convex.

math.AP↗
arXiv2023

Metric viscosity solutions and distance-like functions on the Wasserstein space

Viscosity solutions to the eikonal equation |Du|g = 1, known to be exactly distance-like functions, on a non-compact complete Riemannian manifold (M,g) are crucial for understanding the underlying geometric and topological properties. In this work, we explore metric viscosity solutions, distance-like functions and their relationship on a metric space, especially on the Wasserstein space Pp(X) where X is a complete, separable, locally compact and non-compact geodesic space. Meanwhile, we provide two distinct ways to construct (strong) metric viscosity solutions on Pp(X) and study their properties.

math.AP↗
arXiv2023

Viscosity solutions to a Cauchy type problem for timelike Lorentzian eikonal equation

In this paper, we propose a Cauchy type problem to the timelike Lorentzian eikonal equation on a globally hyperbolic space-time. For this equation, as the value of the solution on a Cauchy surface is known, we prove the existence of viscosity solutions on the past set (future set) of the Cauchy surface. Furthermore, when the time orientation of viscosity solution is consistent, the uniqueness and stability of viscosity solutions are also obtained.

math.AP↗
arXiv2022

Design and analysis of computer experiments with both numeral and distribution inputs

Nowadays stochastic computer simulations with both numeral and distribution inputs are widely used to mimic complex systems which contain a great deal of uncertainty. This paper studies the design and analysis issues of such computer experiments. First, we provide preliminary results concerning the Wasserstein distance in probability measure spaces. To handle the product space of the Euclidean space and the probability measure space, we prove that, through the mapping from a point in the Euclidean space to the mass probability measure at this point, the Euclidean space can be isomorphic to the subset of the probability measure space, which consists of all the mass measures, with respect to the Wasserstein distance. Therefore, the product space can be viewed as a product probability measure space. We derive formulas of the Wasserstein distance between two components of this product probability measure space. Second, we use the above results to construct Wasserstein distance-based space-filling criteria in the product space of the Euclidean space and the probability measure space. A class of optimal Latin hypercube-type designs in this product space are proposed. Third, we present a Wasserstein distance-based Gaussian process model to analyze data from computer experiments with both numeral and distribution inputs. Numerical examples and real applications to a metro simulation are presented to show the effectiveness of our methods.

stat.ME↗
arXiv2022

Point-assigned distance-like functions on non-compact geodesic spaces

On a complete, connected, locally compact, non-compact geodesic space $(X,d)$, we assign each compact set a distance-like function. With the help of these functions, we obtain a pseudo-metric on the space of (non-empty) compact subsets of $X$ which is less than the Hausdorff distance. The quotient metric space is closely related to the large scale geometry of the ambient metric space. In particular, we study both extreme cases --the pseudo-metric either vanishes or equals the original distance. The compactness of the level sets as well as stability under the Gromov-Hausdorff topology of such dl-functions are also investigated. As an application, we also give a representation formula of any distance-like function in terms of the singleton-assigned distance-like functions defined here.

math.DS↗
arXiv2020

Multitransition solutions for a generalized Frenkel-Kontorova model

We study a generalized Frenkel-Kontorova model. Using minimal and Birkhoff solutions as building blocks, we construct a lot of homoclinic solutions and heteroclinic solutions for this generalized Frenkel-Kontorova model under gap conditions. These new solutions are not minimal and Birkhoff any more. We use constrained minimization method to prove our results.

math.DS↗
arXiv2019

Busemann functions on the Wasserstein space

We study rays and co-rays in the Wasserstein space $P_p(\mathcal{X})$ ($p > 1$) whose ambient space $\mathcal{X}$ is a complete, separable, non-compact, locally compact length space. We show that rays in the Wasserstein space can be represented as probability measures concentrated on the set of rays in the ambient space. We show the existence of co-rays for any prescribed initial probability measure. We introduce Busemann functions on the Wasserstein space and show that co-rays are negative gradient lines in some sense.

math.DS↗
arXiv2019

Heteroclinic solutions for a generalized Frenkel-Kontorova model by minimization methods of Rabinowitz and Stredulinsky

We study heteroclinic solutions of a generalized Frenkel-Kontorova model. Using the methods of Rabinowitz and Stredulinsky, we prove that if the rotation vector of the configuration is rational and if there is an adjacent pair of periodic configurations, then there is a solution that is heteroclinic in one fixed direction and periodic in other directions. Furthermore, if the above heteroclinic solutions have an adjacent pair, then there is a solution that is heteroclinic in two directions and periodic in other directions. The procedure can be repeated to produce more complex solutions. Thus we obtain a variational construction for these minimal and Birkhoff solutions.

math.DS↗
arXiv2018

On class A Lorentzian 2-tori with poles II: Foliations by timelike lines

We show that if $(\mathbb{T}^{2},g)$ is a class A Lorentzian 2-torus with timelike poles, then there exists a Lipschitz foliation by complete future-directed timelike geodesics with any pre-assigned asymptotic direction in the interior of the stable time cone. This is done by constructing certain $C^{1,1}$ solutions to the equation $g(\nabla u,\nabla u)=-1$ on the Abelian cover $(\mathbb{R}^{2},g)$.

math.DS↗
arXiv2018

On class A Lorentzian 2-tori with poles I: Closed geodesics pass through poles

In this paper, by studying certain isometries on globally hyperbolic planes, we prove that if $p$ is a timelike pole on a class A Lorentzian 2-torus, then there exists a closed timelike geodesic passing through $p$ with any preassigned free homotopy class in the interior of the stable time cone. We also show a non-rigid result when timelike poles appear.

math.DS↗
arXiv2016

Global viscosity solutions for eikonal equations on class A Lorentzian 2-tori

On the Abelian cover $(\mathbb{R}^{2},g)$ of a class A Lorentzian 2-torus $(\mathbb{T}^{2},g)$, we showed the existence of global viscosity solutions to the eikonal equation $$ g(\nabla u,\nabla u)=-1 $$ associated to those homologies in the interior of the homology cone. Some other related dynamical properties are also considered. As an application of the main results, we study the differentiability of the unit sphere of the stable time separation associated to the class A Lorentzian 2-torus.

math.DS↗
arXiv2014

Viscosity solutions, ends and ideal boundary

On a smooth, non-compact, complete, boundaryless, connected Riemannian manifold $(M,g)$, there are three kinds of objects that have been studied extensively: $\bullet$ Viscosity solutions to the Hamilton-Jacobi equation determined by the Riemannian metric; $\bullet$ Ends introduced by Freudenthal and more general other remainders from compactification theory; $\bullet$ Various kinds of ideal boundaries introduced by Gromov. In this paper, we will present some initial relationship among these three kinds of objects and some related topics are also considered.

math.DS↗
arXiv2013

Busemann functions and barrier functions

We show that Busemann functions on a smooth, non-compact, complete, boundaryless, connected Riemannian manifold are viscosity solutions with respect to the Hamilton-Jacobi equation determined by the Riemannian metric and consequently they are locally semi-concave with linear modulus. We also analysis the structure of singularity sets of Busemann functions. Moreover we study barrier functions, which are analogues to Mather's barrier functions in Mather theory, and provide some fundamental properties. Based on barrier functions, we could define some relations on the set of lines and thus classify them. We also discuss some initial relations with the ideal boundary of the Riemannian manifold.

math.DS↗
arXiv2013

Locally Lipschitz graph property for lines

On a non-compact, smooth, connected, boundaryless, complete Riemannian manifold $(M,g)$, one can define its ideal boundary by rays (or equivalently, Busemann functions). From the viewpoint of Mather theory, boundary elements could be regarded as the static classes of Aubry sets, and thus lines should be think as the semi-statics curves connecting different static classes. In Mather theory, one core property is Lipschitz graph property for Aubry sets and for some kind of semi-static curves. In this article, we prove a such kind of result for a set of lines which connect the same pair of boundary elements.

math.DS↗
arXiv2010

On commuting Tonelli Hamiltonians: Autonomous case

We show that the Aubry sets, the Mañé sets, Mather's barrier functions are the same for two commuting autonomous Tonelli Hamiltonians. We also show the quasi-linearity of $α$-functions from the dynamical point of view and the existence of common $C^{1,1}$ critical subsolution for their associated Hamilton-Jacobi equations.

math.DS↗