Editing Wiener process (section)

==== Local time ====
The image of the [[Lebesgue measure]] on [0, ''t''] under the map ''w'' (the [[pushforward measure]]) has a density {{math|''L''<sub>''t''</sub>}}.  Thus,
<math display="block"> \int_0^t f(w(s)) \, \mathrm{d}s = \int_{-\infty}^{+\infty} f(x) L_t(x) \, \mathrm{d}x </math>
for a wide class of functions ''f'' (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density ''L<sub>t</sub>'' is (more exactly, can and will be chosen to be) continuous. The number ''L<sub>t</sub>''(''x'') is called the [[local time (mathematics)|local time]] at ''x'' of ''w'' on [0, ''t'']. It is strictly positive for all ''x'' of the interval (''a'', ''b'') where ''a'' and ''b'' are the least and the greatest value of ''w'' on [0, ''t''], respectively. (For ''x'' outside this interval the local time evidently vanishes.) Treated as a function of two variables ''x'' and ''t'', the local time is still continuous. Treated as a function of ''t'' (while ''x'' is fixed), the local time is a [[singular function]] corresponding to a [[atom (measure theory)|nonatomic]] measure on the set of zeros of ''w''.

These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however,  the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.