On 1-point densities for Arratia flows with drift

Abstract

We show that if drift coefficients of Arratia flows converge in $L_1(\mathbb{R})$ or $L_{\mathrm{\infty }}(\mathbb{R})$ then the 1-point densities associated with these flows converge to the density for the flow with the limit drift. The main result is proven by providing a representation of the probability of coalescence in the flow as a solution to a PDE and applying the perturbation theory. 

Keywords:

• Brownian web
• Arratia flow
• point process
• point density

AMS CLASSIFICATIONS:

• Primary 60H10
• Secondary 60G55
• 35C10

Acknowledgements

The authors are grateful to the anonymous referee for the helpful comments which significantly improved the presentation.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The authors are grateful to a colleague for providing the reference [2].

2 The statement of [19, Theorem 2.7] contains a misprint: $Q(t,x)$ should be equal $\mathrm{P}({\tau }_A>t)$.