Feller property of the multiplicative coalescent with linear deletion

Abstract

We modify the definition of Aldous’ multiplicative coalescent process (Ann. Probab. 25 (1997) 812–854) and introduce the multiplicative coalescent with linear deletion (MCLD). A state of this process is a square-summable decreasing sequence of cluster sizes. Pairs of clusters merge with a rate equal to the product of their sizes and clusters are deleted with a rate linearly proportional to their size. We prove that the MCLD is a Feller process. This result is a key ingredient in the description of scaling limits of the evolution of component sizes of the mean field frozen percolation model (J. Stat. Phys. 137 (2009) 459–499) and the so-called rigid representation of such scaling limits (Electron. J. Probab. To appear).