{W(x, y), x>=0, y>=0} be a Wiener process and let [eta](u, (x, y)) be its local time. The continuity of [eta] in (x, y) is investigated, i.e., an upper estimate of the process [eta]([mu], [x, x + [alpha]) - [y, y + [beta])) is given when [alpha][beta] is small.

The area of the largest circle around the origin completely covered by a simple symmetric plane random walk is investigated.