The inclusion process: duality and correlation inequalities

We prove a comparison inequality between a system of independent random walkers and a system of random walkers which interact by attracting each other -a process which we call here the symmetric inclusion process (SIP). As an application, correlation inequalities for the SIP, as well as for a model of heat conduction, the so-called Brownian momentum process, are obtained. These inequalities are counterparts of the inequalities (in the opposite direction) for the symmetric exclusion process, showing that the SIP is a natural bosonic analogue of the symmetric exclusion process (which is fermionic). We discuss stationary measures of the SIP, and an asymmetric version that has the same stationary probability measures, as well as infinite non-translation invariant reversible measures. Finally, we consider a boundary driven version of the SIP for which we prove duality and correlation inequalities.