Variational and optimal control representations of conditioned and driven processes

Abstract

We have shown recently that a Markov process conditioned on rare events involving timeintegrated random variables can be described in the long-time limit by an effective Markov process, called the driven process, which is given mathematically by a generalization of Doob's h-transform. We show here that this driven process can be represented in two other ways: first, as a process satisfying various variational principles involving large deviation functions and relative entropies and, second, as an optimal stochastic control process minimizing a cost function also related to large deviation functions. These interpretations of the driven process generalize and unify many previous results on maximum entropy approaches to nonequilibrium systems, spectral characterizations of positive operators, and control approaches to large deviation theory. They also lead, as briefly discussed, to new methods for analytically or numerically approximating large deviation functions.

Large deviations of the current for driven periodic diffusions

“…For the rest of the paper, it is useful to note that the large deviation functions λ(k) and I(j) can also be obtained in a very different way via optimization problems derived in [27] (see also [10,11,52]). For the SCGF, the optimization to solve is…”

“…To be more precise, it is the process that is equivalent to the conditioned process in the long-time limit, in the same way that the canonical ensemble is equivalent to the microcanonical ensemble in the infinite-volume limit [28]. In fact, equivalence is achieved similarly to equilibrium by relating the constant current j of the conditioned (microcanonical) process to the "temperature" k of the auxiliary (canonical) process according to [26] I (j) = k (27) or, equivalently,…”

“…Our goal here is to complete these studies by investigating the large deviation functions characterizing the fluctuations of J T in all noise regimes, and by constructing the auxiliary process, also known as the biased or driven process, describing the diffusion conditionally on observing a current fluctuation J T = j far from the mean current J T . This process is physically important as it describes, by means of a modified stochastic process, how fluctuations of the current or any time-integrated observable in general are created in time [23][24][25][26][27][28]. This effective description of fluctuations was illustrated recently in the context of interacting particle systems [29][30][31][32][33], diffusions [34][35][36], and quantum systems [37][38][39][40][41].…”

Symmetries and Geometrical Properties of Dynamical Fluctuations in Molecular Dynamics

“…As a final point, we note that the rate functions for path probabilities (57) and (69) were derived by considering many copies of our system, but the same formulae also govern large deviations at level 2.5 [17,22,32,40,41]. These LDPs involve rare events where an unusual density or current is sustained over a long time period (in a single system).…”

“…At the level of sample paths, b =b. We identify γ(∂G s /∂p i ) as a "control force" [21,22] that realises the required flux J (see also [20]). …”