台大校區機率演講公告Probability Seminars on NTU Campus

We review the development of random matrix theory beginning with Eugene Wigner's pioneering work in the 1960s. Wigner's profound insight into the universality of eigenvalue distributions has had a lasting influence on the field. We also discuss Dyson's Brownian motion, which has emerged as a powerful tool for establishing and understanding Wigner's universality conjecture. Finally, we highlight applications of these ideas to contemporary research problems in graph theory and related areas.

In this talk, we discuss estimates on the heat kernels of discontinuous symmetric Markov processes including ones with jump kernels degenerate at the boundary. Depending on the regions where the parameters belong, the heat kernels estimates have three different forms, two of them are qualitatively different from all previously known heat kernel estimates. We also discuss the processes killed either by a critical potential or upon hitting the boundary. Their heat kernel estimates have the approximate factorization property with survival probabilities decaying as a power of the distance to the boundary, where the power depends on the critical potential.

This talk is based on joint papers with Soobin Cho, Renming Song and Zoran Vondraček.

Following a brief discussion of the Gaussian beta ensemble and the classical LLN of its empirical measures in the fixed and high temperature regimes, we switch to the discrete setting. By using Fourier transforms based on Jack symmetric polynomials, we study discrete N-particle ensembles in the regime where the inverse temperature parameter tends to zero, as the number of particles N tends to infinity. We prove the LLN and characterize the limiting measure in terms of a moment problem. For fixed-time distributions of Markov chains of N non-intersecting particles (discrete versions of the Dyson Brownian motion), the limiting measures are expressed in terms of the zeros of certain entire functions or the eigenvalues of Jacobi operators. The talk is based on joint work with Maciej Dolega.

Proposed by Kardar, Parisi, and Zhang in 1986, the KPZ equation is a stochastic PDE central to describing a large class of models of random growth and beyond. Among the many interesting properties that the KPZ equation exhibits is its stochastic integrability. In this talk, I will review one way of understanding the integrability via a stochastic/quantum many-body system, the delta Bose gas. I will explain how the latter helps understand the KPZ equation, particularly its tail probabilities and limit shapes.

Recently, there has been an attempt to rewrite the free energy of spin glass models as a solution to a Hamilton-Jacobi equation. In the talk, I will overview the efforts and the results that several people have achieved. Then, I will talk about a joint work with Alexander Alban and Justin Ko where we try to apply this framework to the branching random walk setting. In the weak correlation regime, where the CREM's covariance function A is bounded above by the identity, we show that the free energy is given by the Hopf formula. In contrast, we provide an example outside of the weak regime, where A is a two-segment linear function exceeding the identity. Here, the Hamilton-Jacobi framework no longer applies, and a formal application of the associated variational principle fails to yield the correct free energy.

Quantum disordered system is one of the most important topics in mathematical physics, and related to various interesting phenomena, such as Anderson localization, quantum Hall effect, topological insulator, etc. A typical characteristics is densely distributed point spectrum with exponentially localized eigenfunctions, but recently its statistical properties are drawing much attention. In this talk, we first overview the developments on the 1-dimensional decaying models which exhibits continuous transitions for various properties, and then discuss some of the recent topics :
(1) $d$-dimensional decaying models : we have a phase transition on the spectrum, depending on the tail of the on-site distribution. Extremal value statistics is also observed.
(2) models with critical energies : some special models (e.g., polymer-model, mosaic-model) have finite set of critical energies embedded in the localized regime, because of which they have non-vanishing transport. We see a \"sharp\" transition for the local eigenvalue statistics: from clock to Poisson with no intermediate ones.
(3) $H_{2|2}$-model : this is a 1-dimensional random Schr¨odinger operator with explicit connection to VRJP, a RWRE, and non-linear $\\sigma$-model. In this model, transition of spectral properties is linked to the recurrence-transience of RWRE, and behavior of correlation functions of the $\\sigma$-model.

Consider an integro-differential operator L in the Euclidean space that has both diffusive and non-local parts. We will show under some mild conditions, there is a Feller process having strong Feller property associated with it. We then discuss the solvability of the Dirichlet problem for L and its probabilistic representation.

In this short course on diffusion processes, we will first go over basic facts on Brownian motion, stochastic integral and Ito's formula, and then introduce diffusion processes as solutions to stochastic differential equations driven by Brownian motion. The last part of this course is an introduction of recent work on threshold diffusions, which solve the stochastic differential equations with step-function drift and diffusion coefficients.

Outline & Descriptions
Lecture 1: Brownian motion as a Gaussian process, as a Markov process and as a martingale, the zero set of Brownian motion;
Lecture 2: stochastic integral, Ito’s formula and stochastic differential equations;
Lecture 3: basics of diffusion processes, threshold diffusion and its exit problem, potential measure and transition density.

References
1. Richard Durrett (1996): Stochastic Calculus: A Practical Introduction. CRC Press.
2. Gregory F. Lawler (2006): Introduction to Stochastic Processes. Chapman & Hall/CRC.

We consider first passage percolation with identical and independent weight distributions. In this talk, we study the upper tail large deviations under a tail assumption on the distribution. We prove that the corresponding rate function is described by the so-called discrete p-Capacity, and we study its asymptotic. The talk is based on joint work with Clement Cosco.

Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This is the last talk of the series of lectures. In this talk, we prove convergence of the lace expansion for the nearest-neighbor model in sufficiently high dimension, and for sufficiently spread-out model in dimensions d > 4. Furthermore, we discuss further results for SAW in high dimensions.

Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This is the second talk of the series of lectures. In this talk we focus on the diagrammatic estimates of the lace expansion for SAW, and the random-walk estimates.

Self-avoiding walk (SAW) is a walk that each path does not visit the same point more than once. It is a model of fundamental interest in combinatorics, probability theory, statistical physics and polymer chemistry. It is certainly non-Markovian. These features makes the subject difficult, and many of the central problems remain unsolved.
This series of lectures will be divided into three talks. The first talk, we first give an overview the SAW and review the lace expansion.

The lace expansion is one of the powerful tools for investigating critical phenomena. It has succeeded in obtaining asymptotic expansions for the critical point in several models, e.g., the self-avoiding walk, ordinary/oriented percolation, the contact process, etc. One of our aims is to obtain such an asymptotic expansion for the quantum Ising model, which is one of the models of ferromagnetism. In this model, Spin configurations are randomly realized by the Hamiltonian operator (energy), which involves the z-axis and x-axis Pauli matrices. In other words, the quantum Ising model is defined by applying a transverse field to the classical Ising model. Due to the transverse field, the quantum Ising model exhibits a different type of phase transition from the classical case.

In this talk, I will show the derivation of a lace expansion for the quantum Ising model and upper bounds on the expansion coefficients. The derivation is based on the random current representation [Björnberg and Grimmett (2009) <i>J. Stat. Phys.</i>] [Crawford and Ioffe (2010) <i>Commun. Math. Phys.</i>] on space-time. This representation provides the connectivity between two vertices in space-time, which is similar to percolation models. It helps us to derive the lace expansion.

This talk is based on joint work with Akira Sakai (Hokkaido University, Japan).

We consider an optimal consumption and investment problem with delay under a linear Gaussian stochastic factor model. A linear Gaussian stochastic factor model is a stochastic factor model in which the mean returns of risky assets depend linearly on underlying economic factors that are formulated as the solutions of linear stochastic differential equations. We consider the performance-related capital inflow/outflow, which implies that the wealth process is modeled by a stochastic differential delay equation. We also treat the partial information case where the investor cannot observe the factor process and can use only past information about risky assets. Under this setting, the investor tries to maximize the finite horizon discounted expected HARA utility of consumption, the terminal wealth, and the average wealth. A pair of forward-backward stochastic differential equations derived via the stochastic maximum principle have an explicit solution that can be obtained by solving a time-inhomogeneous Riccati differential equation. Thus, the optimal strategy and the optimal value can be obtained explicitly.

A log-optimal portfolio is any portfolio that maximizes the expected logarithmic growth (ELG) of an investor’s wealth, which typically assumes prior knowledge of the true return distribution. However, in practice, return distributions are often ambiguous; i.e., the true distribution is unknown, making this problem challenging to solve. This paper proposes a supporting hyperplane approximation approach, reformulating a class of distributional robust logoptimal portfolio problems with polyhedron ambiguity sets into tractable robust linear programs. An efficient algorithm is presented to determine the optimal number of hyperplanes. Additionally, to adapt to the constantly changing market, we propose an online trading algorithm using a sliding window approach to solve a sequence of robust linear programs, offering significant computational advantages. The effectiveness of the proposed approach is supported by empirical studies using historical stock price data.

We study Merton's expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown. The agent under consideration is a price taker who has access only to the stock and factor value processes and the instantaneous volatility. We propose an auxiliary problem in which the agent can invoke policy randomization according to a specific class of Gaussian distributions, and prove that the mean of its optimal Gaussian policy solves the original Merton problem. With randomized policies, we are in the realm of continuous-time reinforcement learning (RL) recently developed in Wang et al. (2020) and Jia and Zhou (2022a,b, 2023), enabling us to solve the auxiliary problem in a data-driven way without having to estimate the model primitives. Specifically, we establish a policy improvement theorem based on which we design both online and offline actor–critic RL algorithms for learning Merton’s strategies. A key insight from this study is that RL in general and policy randomization in particular are useful beyond the purpose for exploration – they can be employed as a technical tool to solve a problem that cannot be otherwise solved by mere deterministic policies. At last, we carry out both simulation and empirical studies in a stochastic volatility environment to demonstrate the decisive outperformance of the devised RL algorithms in comparison to the conventional model-based, plug-in method. Joint work with Min Dai, Yuchao Dong and Yanwei Jia.