Research Interests
Probability theory and its applications (e.g. OR, Machine Learning/Deep Learning, stochastic modeling, etc.); with differential geometry methods and their interactions and applications to probability theory.
Working Papers and Projects
Sticky Brownian Motion and the Submartingale Problem. Ziran Liu, 2024.
Abstract: We study a type of Brownian motion with drift constrained to a wedge in the plane. The feature is that this Brownian motion reflects on the boundary of the wedge but behaves as a sticky Brownian motion at the vertex of the wedge. As far as the literature shows, this type of Brownian motion has not been studied before, and we study it by formulating it into a kind of submartingale problem. This is also an extension of the question asked by Prof. Varadhan on the author's thesis defense.
Heavy Traffic Analysis of the Coupled Queue Processor System. Ziran Liu, 2024.
Abstract: We study the heavy traffic case of the queueing process generated by the coupled queue processor. This paper establishes the heavy traffic limit theorem for the coupled queue process, with the limiting process established in the paper by the author and Lakner and Reed.
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PhD Thesis Research
Studies on reflected Brownian motion, and its applications to geometric invariants [Ph.D. Thesis]. Ziran Liu, 2022.
*This thesis is tailored into my job market paper (the following paper on this page) titled "Reflected Brownian motion with drift in a wedge", accepted for publication at Queueing Systems: Theory and Applications (QUESTA).
An intriguing but classical question in probability theory and Operations Research (OR) is the "heavy-traffic" limit process (usually a diffusion process) of some featured queueing processes (eg. a birth-death process), also known as the diffusion approximation in literature. Mathematically, it is to study the continuous limiting process taking limit in a "right" sense of some discrete-stated processes. It can provide insight into (the global, meaning extremely long-time) behavior of a queueing process, by studying its (functional) limit in a "right" sense (i.e., there can be multiple ways of taking limit in a suitable probabilistic or even deterministic sense).
In particular, the non-negativeness of the pre-limit queueing processes leads to the most prominent feature of its limiting process being "constrained" in a domain instead of moving freely in the whole space. A classical example for this phenomenon in most of the literature is the limiting processes being constrained in the first orthant in the Euclidean space. Therefore the limiting process has to be "reflected" at the boundary of a domain, thus we call this reflected processed. (The word "reflect" is used for historical reasons, although it only means the process is constrained in the domain and blocked by the boundary from going out, instead of the usual meaning of "reflection" in physics.)
Contribution: This PhD research investigates such kind of limiting processes but in a more general and non-traditional setting, i.e., with the state space of a wedge, and with oblique reflection, compared to the traditional case of the state space being the first orthant. The problem under investigation is formulated as the submartingale problem with drift term in a wedge, which is a way of the mathematical formulation of the reflected Brownian motion with drift. This is also a generalization of the work by Varadhan and Williams in 1985. But a key difference in this Ph.D. research is to include the non-zero drift term. Methodologically, the drift makes the explicit solution to a concerned Laplace equation (a key ingredient in the method in Varadhan and Williams in 1985) not exist, nonetheless it makes the solution's analytic property become unclear and varies from case to case. Physically, the drift can essentially change the long-time behavior of the reflected process, which can beneficially furthermore make the stationary distribution exist. Here we prove the existence and uniqueness properties of this submartingale problem with drift term, strong Markov property, and two types of different Feller properties which are stronger than that in Varadhan and Williams in 1985 even when drift is zero. As well, we provide quantitative properties such as the hitting probability of the vertex of the wedge for the absorption processes and the connection between this solution and the Skorokhod problem.
As a fact, the reflected process under research, i.e., the reflected Brownian motion (RBM), leads to interesting applications in differential geometry. The well-known Gauss-Bonnet formula for the Euler characteristic generalizes to the Gauss-Bonnet-Chern formula for manifolds without boundary. It is noticeable that the discussion of the Gauss-Bonnet-Chern formula with boundary is not as well-populated as the case without the boundary, one reason is the "correction term" on the boundary is not formulated simply. However, this is interesting because RBM provides a new way to compute the boundary term in the Gauss-Bonnet-Chern formula with boundary, as a reference one can see Du and Hsu 2022. This leads to a natural question, also in the scope of the index theorems, that whether there is a probabilistic understanding of the Aityah-Patodi-Singer (APS) index theorem for Dirac operators on spin manifolds with boundary. The question is both tricky and interesting because the "correction term" in the APS is eta-invariant, which is globally defined by the spectrum of the underlying Dirac operator, as opposed to a geometric quantity purely defined on the boundary. If one can see the connection between a proper reflected process and the way of computing the eta-invariant, it will, on one hand, give an innovative understanding to the eta-invariant but also can possibly reveal some interesting "cancellation phenomenon", vividly speaking it is like the contribution of the Dirac operator to the eta-invariant seems like canceled inside the manifold by itself, with the contribution on the boundary remained only. How to understand this possible "cancellation"? This is the core idea of the second part of the thesis research, the key idea is centered on the use of the reflected process and its related probabilistic behavior. An extensive study has been carried out and related intermediate understanding has been exhibited, the focus on the original question also leads to subquestions. Some futuristic research problems are also presented.
Reflected Brownian motion with drift in a wedge. Peter Lakner, Ziran Liu, Josh Reed, published at Queueing Systems: Theory and Applications (QUESTA).
*This is the published version of the first half of my thesis research.
This paper studies the reflecting Brownian motion (RBM) with drift constrained to a wedge in the plane. Mathematically, it is formulated in the paper as the submartingale problem with drift term and oblique reflection in a wedge-shaped area. The first set of results provides necessary and sufficient conditions for the existence and uniqueness of a solution to the corresponding submartingale problem with drift. Then we show that its solution possesses strong Markov and Feller properties. Quantitatively, as the second part of the results, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for the existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.
A Note on Hausdorff Multifractal Spectrum of Log-correlated Gaussian Multiplicative Chaos. Ziran Liu, 2020.
In the populated notes on Gaussian multiplicative chaos and Liouville Quantum Gravity by Rhodes and Vargas 2016, the notes are based on lectures given by the second author in Les Houches's summer school. Interestingly, "Theorem 2.6" from the notes on multifractal formalism has been claimed without being proved. However, Theorem 2.6 had not appeared in previous papers or books with proof. Here, we give complete proof to this theorem, not using the possible suggested idea in Rhodes and Vargas 2014, we essentially use the power-law spectrum of the p-moment of the Gaussian multiplicative chaos (GMC).
Assortment Optimization: A Near Linear Algorithm for a Type of Choice Modeling Problem. Ziran Liu, Preprint, 2020.
In Operation Research (OR) with Economics, a significant problem is maximizing revenue by optimizing the set of products offered to customers in different scenarios. This research considers this problem in the Choice Modeling setting, especially for the mixed multinomial logit (MMNL) model. Mathematically, the underlying is a combinatorial problem to find a subset with at most C elements out of a universal set with N elements to maximize the expected revenue of the set of products offered to the customer. The hardness of such an optimization is apparent; it faces the curse of dimensionality for C near N, i.e., the number of choices of subsets is getting exponential. Here, we investigate the algorithm for this choice modeling problem. On one hand, like in many other algorithmic problems, the idea of local search shows tremendous power; on the other hand, we will also center on a local search algorithm for such a choice modeling problem, called ADXOpt from Jagabathula 2014.
In this research, we study the guarantee of the ADXOpt problem for the MMNL model and give a bound that depends on the cardinality of the subset offered but is near sublinear concerning a constant fixed for each determined MMNL model (defined in Jagabathula 2014). The contribution is that this bound compares to the counterpart bound in Jagabathula 2014 but exponentially decays.
Gaussian Processes and Kolmogorv-Wiener Problem. [M.S. Thesis] Ziran Liu, 2016.
*This is my thesis for my M.S. from NYU Courant Institute of Mathematical Sciences, under the supervision of Prof. Henry McKean.
In the book by McKean and Dym 1976, the authors investigate the prediction problems for the Gaussian process by using the Hardy function and the analytical structure of the Gaussian process itself. This thesis can be considered an investigation to answer follow-up questions using the Hardy function to predict the Gaussian process.
A Lichnerowicz vanishing theorem on non-compact spin manifolds with a type of Lie group action. [M.S. Thesis] Ziran Liu, Preprint, 2014.
*This is my thesis for my M.S. from Chern Institute of Mathematics at Nankai University, China. The thesis boils down to a short note, including the main theorem. "A Lichnerowicz vanishing theorem for proper cocompact actions." Ziran Liu, 2013. arXiv:1310.4903.
This thesis (as the main theorem in the note by Ziran Liu, 2013 as above) establishes and proves the Lichnerowicz type vanishing theorem for the Mathai-Zhang index when the action Lie group is unimodular. It works as a transverse index on a singular foliation from group action compared to the longitudinal index considered in Tang, Willett, and Yao 2016.
Interestingly, the method in this work was also deployed by Zhang 2015 which generalizes to the Lie group, which is not unimodular.
Concisely speaking, the Lichnerowicz formula is the equation connecting the square of the Dirac operator and the sum of Laplacian and the quarter of the scalar curvature of the underlying Riemannian manifold. However, the Atiyah-Singer theorem for Dirac operators shows that the Fredholm index of the Dirac operator equals the A-hat genus of the manifold. Combining these two points, one can see that the A-hat genus is an obstruction for the underlying Riemannian manifold to have positive scalar curvature. Vice versa, if the manifold carries a positive scalar curvature, the "obstruction" is broken so that the A-hat genus or the Fredholm index of the Dirac operator vanishes. The work in this thesis can be considered as research in this line.