Steven Orey

Gaussian sample functions and the Hausdorff dimension of level crossings

SummaryLet Xt be a real Gaussian process with stationary increments, mean 0, σt2=E[(Xs+t−Xs)2] If σ σt2 behaves like tα as t ↺0, 0<α<1, the graph of a.e. sample function will have Hausdorff dimension 2 -α. This leads one to feel that the set of zeros of Xt should have Hausdorff dimension 1 -α. This is shown to be true provided the process is stationary and satisfies additional assumptions.

Small random perturbation of dynamical systems with reflecting boundary

Consider a diffusion process in R d satisfying the stochastic differential equation .

Minimizing or maximizing the expected time to reach zero

We treat the following control problems: the process

ON THE RANGE OF RANDOM WALK

X_1 (t)

On

with Values in the interval

\omega

( { - \infty ,0} ]

-Consistency and Related Properties

(or

LARGE DEVIATIONS IN ERGODIC THEORY

[ {0,\infty } )

Reaching zero rapidly

) is given by the stochastic differential equation \[dX_1 (t) = \mu (t)dt + \sigma (t)dW_t ,\quad X_1 (0) = x_1 \] where the nonanticipative controls

Stationary solutions for linear systems with additive noise

\mu