Entropy, action and the onsager-machlup function

Abstract

A generalization of these results to multidimensional processes, processes on manifolds, non-autonomous processes and various boundary conditions is possible. Some results are given in n Moreover, function space techniques may be helpful for the understanding of the transformation of realizations of the driving process — the Wiener process — into the resulting trajectories of the process under consideration.

The fundamental idea of a deterministic description of an irreversible process by means of properties of its realizations depends on the choice of an absolutely continuous m.p.p. in order to obtain a connection to the conventional description of dynamics by differential equations. Although μ(C'(x_i))=o for all diffusion processes², the necessity of x ɛ C'(x_i) for quasitranslation invariance determines the basis of a deterministic description which is analogous to canonical classical mechanics. The irreversibility of these dynamics is reflected mathematically by the fact that the m.p.p. is determined by a boundary-value problem and not by an initial value problem.

From these considerations a comment to the formal approach to OMF theory seems appropriate. Formal application of an infinite-dimensional Lebesgue measure by means of an iteration of the Chapman-Kolmo gorov equation leads to a path integral description of the t.p.d. It is not granted that the integration prescription involved, the sequential Wiener integral or generalizations of it, defines a measure at all, and if it does it can be proved that it may differ from the Wiener integral defined by means of the induced measure11. These properties result form the fact that the induced measure is constructed according to Lebesgue theory whereas the sequential integral is, by its definition, a linear functional on a non locally compact space for which Riesz's representation theorem does not apply. Due to the insertion of a (quite arbitrary) short-time approximation of the t.p.d. into the sequential integral the Lagrangian thus defined becomes prescription dependent¹².

The procedure presented here is based on the fact that a description-dependent representation is avoided from the beginning. This allows for an unambiguous derivation of the OMF and a self-evident probabilistic interpretation of the OMF and the m.p.p. which is not granted otherwise.