Harnack inequality for mean-field stochastic differential equations

Buckdahn et al. (2009b) introduced a mean-field stochastic differential equation to study the backward stochastic differential equation. The objective of the present paper is to deepen the investigation of such mean-field stochastic differential equations by studying them in a Brownian motion framework. By constructing a coupling, loglog-Harnack inequality and Harnack inequality with dimension-free are established for such mean-field stochastic differential equations.