Brownian Motion with Singular Time-Dependent Drift

TL;DR

This paper establishes the existence and uniqueness of weak solutions for Brownian motion with singular, time-dependent drift functions belonging to a specific Kato class, expanding understanding of stochastic differential equations with irregular drifts.

Contribution

It proves the well-posedness of SDEs with drifts in the forward-Kato class, a significant extension to cases with singular, time-dependent drifts.

Findings

Unique weak solutions exist under Kato class conditions.

Solutions are valid for all starting points.

The results apply to drifts with singularities in time and space.

Abstract

In this paper we study weak solutions for the following type of stochastic differential equation \[ dX_{t}=dW_{t}+b(t, X_{t})dt, \quad t\ge s, \quad X_{s}=x, \] where $b: [0,\infty) \times \mathbb{R}^{d} \to \mathbb{R}^{d}$ is a measurable drift, $W=(W_{t})_{t \ge 0}$ is a $d$-dimensional Brownian motion and $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$ is the starting point. A solution $X=(X_t)_{t \ge s}$ for the above SDE is called a Brownian motion with time-dependent drift $b$ starting from $(s,x)$. Under the assumption that $|b|$ belongs to the forward-Kato class $\mathcal{F} \mathcal{K}_{d-1}^{\alpha}$ for some $\alpha \in (0,1/2)$, we prove that the above SDE has a unique weak solution for every starting point $(s,x)\in [0,\infty) \times \mathbb{R}^{d}$.