this paper, we show that the answer to the above question is positive. More specifically, we give a canonical construction for semi-pullbacks in the category whose objects are families of Markov processes, with given transition kernels, on Polish spaces and whose morphisms are transition probability preserving surjective continuous maps. One immediate consequence is that the category of probability measures on Polish spaces with measure-preserving continuous maps has semi-pullbacks. Our construction gives semi-pullbacks for various full subcategories, including that of Markov processes on locally compact second countable spaces and also in the larger category where the objects are Markov processes on analytic spaces (i.e. continuous images of Polish spaces) and morphisms are transition probability preserving surjective Borel maps. It also applies to the corresponding categories of ultrametric spaces. Finally, our result also holds in the larger categories with Markov processes which are given by subprobability distributions, i.e. the total probability of transition from a state can be strictly less than one. We now explain the relevance of our result in computer science. The consequences of Semi-pullbacks and Bisimulation 3 our mathematical result in the theory of probabilistic bisimulation has been investigated in (Blute et al., 1997; Desharnais et al., 1998). We will briefly review this here. Following the work of Joyal, Nielsen and Winskel (Joyal et al., 1996) on the notion of bisimulation using open maps, define two objects A and B in a category to be bisimular if there exists an object C and morphisms f : C ! A and g : C ! B, i.e.,

. We study nonequilibrium statistical mechanics in the presence of a thermostat acting by random forces, and propose a formula for the rate of entropy production e(¯) in a state ¯. When ¯ is a natural nonequilibrium steady state we show that e(¯) 0, and sometimes we can prove e(¯) ? 0. Key words and phrases: entropy production, nonequilibrium stationary state, nonequilibrium statistical mechanics, random dynamics, SRB state, thermostat. * IHES (91440 Bures sur Yvette, France), and Math. Dept., Rutgers University (New Brunswick, NJ 08903, USA). 0. Physical introduction. The production of entropy in nonequilibrium statistical mechanics was analyzed in various settings in [19]. Here we continue this analysis by studying the role of a heat bath, and its idealization by random external forces. As discussed in [19], we maintain a system outside of equilibrium by external forces which, in general, do not keep the energy constant. If no precaution is taken, the time evolution is then repre...