“My search for truth is finished at last, I’m going home.” ― Zima Blue

Random Walk (2023)

References: [1]. Random Walk: A Modern Introduction, Gregory F. Lawler and Vlada Limic (primary) [2]. Intersections of Random Walks, Gregory F. Lawler[3]. The Art of Random Walks, András Telcs

Ⅰ.General Definitions and Properties

Ⅱ. Local Central Limit Theorem

Ⅲ.Approximated by BM: Skorokhod Embedding and Dyadic Coupling

Ⅳ.Green’s Functions and Potential Kernels

Ⅴ.Potential Theory: Harmonic Functions, Capacity, Dirichlet Problem, Neumann Problem

Ⅵ. Loop Measures and LERWs: Generating Functions, h-Process and LERWs in Z^d

Ⅶ. Intersection Probablity of RWs

Ⅷ.Specific Kinds Random Walks: One-dimension RW and SRW

Large Deviation Theory (2023)

References: [1]. Large Deviations, S. R. S. Varadhan (primary) [2].A basic introduction to large deviations: Theory, applications, simulations, Hugo Touchette [3]. Master Program: Probability Theory, Claudio Landim

Ⅰ.Large Deviation for i.i.d. Sequence

Ⅱ.General Principles: Properties, Contraction THM, Varadhan THM

Advanced Stochastic Process (2022)

References:[1]Probability Theory: Theory and Examples, Richard Durrett [2]Lecture Notes on Measure-theoretic Probability Theory, Sebastien Roch [3]Sample path properties of Brownian motion, Peter Morters

Ⅰ.Discrete Martingales

Ⅱ.Discrete Time Markov Chain

Ⅶ.Path Property of Brownian Motion

Advanced Probability Thoery (2021)

References:[1] A course in probability theory, Kai Lai Chung [2] Probability Theory: Theory and Examples, Richard Durrett [3] Advanced Probability Theory, Bingyi Jing

Ⅵ.Law of Large Number

Ⅶ.Vague Convergence

Ⅷ.Characteristic Functions

Ⅸ.Central Limit Theorem

Complement: Infinite divisiblility

Complement: Ising Model