  In this paper we present a Doob type maximal inequality for stochastic
processes satisfying the conditional increment control condition. If we assume,
in addition, that the margins of the process have uniform exponential tail
decay, we prove that the supremum of the process decays exponentially in the
same manner. Then we apply this result to the construction of the almost
everywhere stochastic flow to stochastic differential equations with singular
time dependent divergence-free drift.
