# On a maximal inequality and its application to SDEs with singular drift

## TL;DR

This paper establishes a maximal inequality for stochastic processes with specific tail decay properties and applies it to construct stochastic flows for SDEs with singular, divergence-free drifts, advancing understanding of such complex systems.

## Contribution

It introduces a Doob type maximal inequality for processes with conditional increment control and applies it to SDEs with singular drifts, enabling the construction of stochastic flows.

## Key findings

- Supremum of processes decays exponentially under certain conditions
- Maximal inequality holds for processes with exponential tail decay
- Application to stochastic flows with singular drifts

## Abstract

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.