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We study the contact process with stirring on $\mathbb{Z}^d$. In thisprocess, particles occupy vertices of $\mathbb{Z}^d$; each particle dies withrate 1 and generates a new particle at a randomly chosen neighboring vertexwith rate $\lambda$, provided the chosen vertex is empty. Additionally,particles move according to a symmetric exclusion process with rate $N$. Forany $d$ and $N$, there exists $\lambda_c$ such that, when the system startsfrom a single particle, particles go extinct when $\lambda < \lambda_c$ andhave a chance of being present for all times when $\lambda > \lambda_c$.Durrett and Neuhauser proved that $\lambda_c$ converges to 1 as $N$ goes toinfinity, and Konno, Katori and Berezin and Mytnik obtained dimension-dependentasymptotics for this convergence, which are sharp in dimensions 3 and higher.We obtain a lower bound which is new in dimension 2 and also gives the sharpasymptotics in dimensions 3 and higher. Our proof involves an estimate fortwo-type renewal processes which is of independent interest.

Improved asymptotic estimates for the contact process with stirring