  We prove a comparison inequality between a system of independent random
walkers and a system of random walkers which either interact by attracting each
other -- a process which we call here the symmetric inclusion process (SIP) --
or repel each other -- a generalized version of the well-known symmetric
exclusion process. As an application, new correlation inequalities are obtained
for the SIP, as well as for some interacting diffusions which are used as
models of heat conduction, -- the so-called Brownian momentum process, and the
Brownian energy process. 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. Finally, we consider a boundary driven version of
the SIP for which we prove duality and then obtain correlation inequalities.
