Hamiltonian and Lagrangian for the Trajectory of the Empirical Distribution and the Empirical Measure of Markov Processes

Abstract

We compute the Hamiltonian and Lagrangian associated to the large deviations of the trajectory of the empirical distribution for independent Markov processes, and of the empirical measure for translation invariant interacting Markov processes. We treat both the case of jump processes (continuous-time Markov chains and interacting particle systems) as well as diffusion processes. For diffusion processes, the Lagrangian is a quadratic form of the deviation of the trajectory from the Kolmogorov forward equation. In all cases, the Lagrangian can be interpreted as a relative entropy or relative entropy density per unit time.

Dynamical Gibbs–non-Gibbs Transitions in the Curie–Weiss Potts Model in the Regime$$\beta <3$$

“…and γ s is the implicit function y = γ s (x) defined in a neighbourhood of (x, y) = (0, s 3 ) by the degeneracy condition (28). Note that Eq.…”

“…To this point corresponds a degenerate stationary point m that has the same symmetry m 2 = m 3 . We can solve the degeneracy condition (28) in a neighbourhood of m in the form y = γ β,t (x) such that γ β,t (0) is the y-coordinate of m. In α-space in a neighbourhood of α = χ(m, β, t) we can now write the bifurcation set as χ(ϕ −1 β (x, γ β,t (x), 0), β, t). We know that the vertial component…”

“…where γ β,t is obtained by solving the degeneracy condition (28) in the form y = γ β,t (x) in a neighbourhood of the point m. This equation is now dependent on m, β and t, that is, we have one equation and three variables (m is one-dimensional because m 2 = m 3 ). Additionally, since we know that the direction of degeneracy is the x-direction, we have the equation…”

“…Proposition 13 Fix a positive β and positive t and consider the set of solutions m to the degeneracy condition (28) with α = χ(m, β, t).…”

“…In this functional the conditioning provides an additional parameter given by an empirical measure on the second layer. This is more direct than working in the Lagrangian formalism on trajectory space, which would provide additional insights on the nature of competing histories that explain the current state of the system at a discontinuity point [9,20,28,32].…”