# Super-Ricci flows and improved gradient and transport estimates

## TL;DR

This paper extends heat flow and transport estimates to super-Ricci flows, enabling the construction of Brownian motions on evolving metric measure spaces and improving gradient inequalities.

## Contribution

It introduces new transport estimates for heat flow on super-Ricci flows and constructs Brownian motions in time-dependent metric spaces.

## Key findings

- Transport estimates hold for all L^p-Kantorovich distances.
- Brownian motions can be constructed on time-dependent metric measure spaces.
- Refined Bochner's inequality is established under stronger metric assumptions.

## Abstract

We show that the heat flow on super-Ricci flows in the sense of Sturm satisfies transport estimates with respect to every $L^p$-Kantorovich distance, $p\in[1,\infty]$. As an application we construct Brownian motions on time-dependent metric measure spaces and present transport estimates for their trajectories. The proof is inspired by the approach from Savar\'e and Bakry respectively and takes advantage of the self-improvement of the gradient estimates. For this we prove a refined version of Bochner's inequality under strengthened assumptions on the metric.