  We investigate, both analytically and numerically, the kinetic and stochastic
counterpart of the triadic Cantor set. The generator that divides an interval
either into three equal pieces or into three pieces randomly and remove the
middle third is applied to only one interval, picked with probability
proportional to its size, at each generation step in the kinetic and stochastic
Cantor set respectively. We show that the fractal dimension of the kinetic
Cantor set coincides with that of its classical counterpart despite the
apparent differences in the spatial distribution of the intervals. For the
stochastic Cantor set, however, we find that the resulting set has fractal
dimension $d_f=0.56155$ which is less than its classical value $d_f={{\ln
2}\over{\ln 3}}$. Nonetheless, in all three cases we show that the sum of the
$d_f$th power, $d_f$ being the fractal dimension of the respective set, of all
the intervals at all time is equal to one or the size of the initiator $[0,1]$
regardless of whether it is recursive, kinetic or stochastic Cantor set.
Besides, we propose exact algorithms for both the variants which can capture
the complete dynamics described by the rate equation used to solve the
respective model analytically. The perfect agreement between our analytical and
numerical simulation is a clear testament to that.
