Wenpin Tang

The Vervaat transform of Brownian bridges and Brownian motion

For a continuous function f 2 C([0;1]), define the Vervaat transform V (f)(t) := f( (f) +t mod 1) +f(1)1ft+ (f) 1g f( (f)), where (f) corresponds to the first time at which the minimum of f is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.

Tree formulas, mean first passage times and Kemeny's constant of a Markov chain

In this paper, we aim to provide probabilistic and combinatorial insights into tree formulas for the Green function and hitting probabilities of Markov chains on a finite state space. These tree formulas are closely related to loop-erased random walks by Wilsons algorithm for random spanning trees, and to mixing times by the Markov chain tree theorem. Let

The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift

m_{ij}

Patterns in Random Walks and Brownian Motion

be the mean first passage time from

Renewal sequences and record chains related to multiple zeta sums

i

Optimal surviving strategy for drifted Brownian motions with absorption

to

Regenerative random permutations of integers

j

The argmin process of Brownian motion and Brownian extrema of a given length

for an irreducible chain with finite state space

Finite and infinite Mallows ranking models, maximum likelihood estimator, and regeneration

S

One-dependent hard-core processes and colorings of the star graph.

and transition matrix