Some properties of solutions of It\^o equations with drift in $L_{d+1}$

TL;DR

This paper investigates properties of solutions to Itô equations with drifts in $L_{d+1}$, including Green's function summability, resolvent boundedness, and Itô's formula, extending previous work on inhomogeneous Markov processes.

Contribution

It introduces new results on Green's functions, resolvent operators, and Itô's formula for solutions with drifts in $L_{d+1}$, advancing understanding of such stochastic processes.

Findings

Higher summability of Green's functions

Boundedness of resolvent operators in Lebesgue spaces

Establishment of Itô's formula for these processes

Abstract

This paper is a natural continuation of [8], 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 higher summability of Green's functions, boundedness of resolvent operators in Lebesgue spaces, establish It\^o's formula, and so on.