Research

I'm interested in probability theory and its applications, with special focus on various aspects on Markov processes.

2. 2K-B is not Bessel(5) (with Lingfu Zhang)

The concave majorant K of a one dimensional Brownian motion the minimum concave function which dominates B. In light of the famous 2M-B theorem of Pitman, it was recently conjectured by Ouaki and Pitman that $2K-B$ has the same law as the BES(5) process. 

We disprove this conjecture, despite the many evidences such that two processes share many properties such as Brownian scaling, time inversion, qudratic variation, and one-point distribution.  

In particular, we show that in a rare event, $2K-B$ behaves essentially like mixture of $\operatorname{BES}(3)$. We derive a path decomposition of $2K-B$ other properties such as several point distribution in the rare event, which are intractable in the original $2K-B$ as pointed out in \cite{OP}. As a byproduct, we characterize how mixture of $\operatorname{BES}(3)$ would, locally at time $0$, behaves like another one dimensional diffusion, which could be viewed as a partial converse of William's path decomposition. 

(Will be on Arxiv soon. Draft available up to requests.)

1.  The 15 Puzzle Problem (with Robert Hough)

Under the supervision of my advisor Robert Hough, we have shown that a single numbered piece on the n^2-1 puzzle given periodic boundary conditions has a randomized location after order n^4 random moves, and that if the number of random moves tends to infinity compared to n^4, then the number of pieces left in their original position converges to a Poisson(1) distribution. The distribution of pieces on the board converges to uniform in O(n^4 log n) random moves. A preprint of our work is available here. An article about the puzzle has appeared in Quanta. A poster is available here.