Brownian motion at a slow point

Source abstract

If c > 1 c > 1 there are points T ( ω ) T(\omega ) such that the piece of a Brownian path B , X ( t ) = B ( T + t ) − B ( T ) B,X(t) = B(T + t) - B(T) , lies within the square root boundaries ± c t \pm c\sqrt t . We study probabilistic and sample path properties of X X . In particular, we show that X X is an inhomogeneous Markov process satisfying a certain stochastic differential equation, and we analyze the local behaviour of its local time at zero.