ENTROPY DISSIPATION ESTIMATES FOR INHOMOGENEOUS ZERO-RANGE PROCESSES

Introduced by Lu and Yau (Comm. Math. Phys. 156 (1993) 399-433), the martingale decomposition method is a powerful recursive strategy that has produced sharp log-Sobolev inequalities for homogeneous particle systems. However, the intractability of certain covariance terms has so far precluded applications to heterogeneous models. Here we demonstrate that the existence of an appropriate coupling can be exploited to bypass this limitation effortlessly. Our main result is a dimension-free modified log-Sobolev inequality for zero-range processes on the complete graph, under the only requirement that all rate increments lie in a compact subset of (0,infinity). This settles an open problem raised by Caputo and Posta (Probab. Theory Related Fields 139 (2007) 65-87) and reiterated by Caputo, Dai Pra and Posta (Ann. Inst. Henri Poincare Probab. Stat. 45 (2009) 734-753). We believe that our approach is simple enough to be applicable to many systems.

Automated summary

Introduced by Lu and Yau, the martingale decomposition method has proven to be a powerful recursive strategy for producing sharp log-Sobolev inequalities in homogeneous particle systems. By demonstrating the potential for an appropriate coupling, the researchers were able to overcome the difficulties posed by certain covariance terms in heterogeneous models. Their main result is a dimension-free modified log-Sobolev inequality for zero-range processes on the complete graph, which resolves an open problem raised by previous researchers. This approach is believed to be straightforward enough for application to various systems.