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.