the construction of an interacting particle system
An interacting particle system (IPS) is a continuous-time Markov process describing a huge collection of simple components — call them spins or particles — sitting on the sites of a lattice or graph S (typically the integer lattice Z^d) and changing their state at random times, where the rate at which any one site changes depends only on the local configuration around it. It is the dynamic counterpart of equilibrium statistical mechanics: instead of asking what configurations a system of many interacting parts prefers in equilibrium, we watch the parts actually evolve and interact in time. The fundamental difficulty the construction must overcome is that the state space is X = {0,1}^S, the set of all configurations eta assigning 0 or 1 to every site — an uncountably infinite product when S is infinite — so the elementary CTMC theory of Vol I (countable state space, exponential holding times) does not apply, because every site is constantly attempting to flip and there is no first jump.
The standard remedy is to define the process not by its jumps but by its generator. One specifies, for each site x and each configuration eta, a nonnegative rate c(x, eta) at which the coordinate eta(x) flips (a spin-flip system) or, for conservative dynamics, a rate c(x, y, eta) at which the contents of sites x and y are exchanged. From these local rates one writes the formal generator acting on local functions f (functions depending on finitely many coordinates): for a spin-flip system, (L f)(eta) = sum_x c(x, eta) [ f(eta^x) - f(eta) ], where eta^x is eta with the coordinate at x flipped. The central theorem (Liggett, building on Hille-Yosida) states that under a boundedness and finite-range / summable-influence condition on the rates — sup_x sum_y (variation of c(x, .) in coordinate y) finite — the closure of L generates a Feller semigroup P_t = e^(tL) of operators on the space C(X) of continuous functions on the compact space X = {0,1}^S, and hence a unique Feller Markov process eta_t with cadlag paths and the strong Markov property. The configuration space X is compact in the product topology (Tychonoff), which is what makes the analytic machinery work and guarantees that invariant measures exist.
This generator-first construction is the foundation of the whole field, and its honest content is a regularity condition that must not be dropped: if the rates can grow unboundedly with the configuration, the formal generator need not be closable and the process can fail to exist or be non-unique. Two structural features inherited from the construction are exploited throughout the subject. First, the strong Markov property lets one restart the process at random times. Second, many natural systems are monotone (attractive), which together with the basic coupling and graphical (percolation-substructure) representations gives a powerful order structure; and the bilinear pairing of the generator with itself yields duality. The graphical construction — drawing independent Poisson processes of arrows and recovery marks on the space-time lattice and reading off the configuration by following the marks — is an especially vivid alternative that builds the process pathwise from independent Poisson clocks and makes the basic coupling automatic.
On Z, consider stirring: each nearest-neighbour edge {x, x+1} carries an independent Poisson clock of rate 1, and when it rings the two occupation values eta(x) and eta(x+1) are swapped. The rates c(x, x+1, eta) = 1 are bounded and finite-range, so Liggett's theorem builds a Feller process on {0,1}^Z. This is the symmetric exclusion process; the same recipe with flip rates c(x, eta) depending on neighbours gives the voter or contact process.
The same generator template — local rates summed over sites — produces the whole zoo of particle systems; only the rates change.
The construction stands or falls on a regularity condition (bounded, summable-influence rates): without it the formal generator need not be closable and the process can fail to exist or be non-unique. 'Continuous-time on an infinite lattice' does not let you list jumps in order — there is no first jump — which is exactly why the generator, not the jump chain, is the primitive object.