On potentials of It\^o's processes with drift in $L_{d+1}$

TL;DR

This paper investigates properties of Itô's processes with drift in the space $L_{d+1}$, focusing on passage probabilities and Green's function properties, extending previous work on inhomogeneous Markov processes.

Contribution

It introduces new results on passage probabilities and Green's function properties for Itô's processes with drift in $L_{d+1}$, even in the case of constant diffusion.

Findings

Enhanced understanding of passage probabilities through narrow tubes

Higher summability results for Green's functions

Properties hold even for constant diffusion

Abstract

This paper is a natural continuation of \cite{Kr_20_2}, where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb{R}^{d+1})$. Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Green's functions, and so on. The results seem to be new even if the diffusion is constant.