For a class of stationary Markov-dependent sequences
(A_n,B_n)
in R^2, we consider the random linear recursion S_n=A_n+B_n
S_{n-1}, n \in \zz, and show that the distribution tail of its
stationary solution has a power law decay.
There is a rich physical literature on polymer dynamics which presents a number of fascinating challenges for mathematicians. We model thermal fluctuation of a polymer in solvent as a curve or loop obeying a stochastic partial differential equation (SPDE). The simplest instance is the so-called Rouse model which is an infinite dimensional Ornstein-Uhlenbeck process satisfying a linear SPDE. We'll review the Rouse model and then describe recent results (a) of Seung Lee on an SPDE for the Rouse model in a half-space with reflecting boundary conditions; and (b) of Scott McKinley on an SPDE model of the hydrodynamic interaction.
We present a coupling of the 1-dimensional Ornstein-Uhlenbeck process with an i.i.d. sequence.
We then apply this coupling to resolve two conjectures of Darling and Erd\H{o}s (1956).
Interestingly enough, we prove one and disprove the other conjecture. [This is joint work with David Levin.]
Time-permitting, we may use the ideas of this talk to describe precisely the rate of convergence in the
classical law of the iterated logarithm of Khintchine for Brownian motion (1933).
[This portion is joint work with David Levin and Zhan Shi, and has recently appeared in
the Electr. Comm. of Probab. (2005)]
