Jeffreys Divergence and Generalized Fisher Information Measures on Fokker–Planck Space–Time Random Field

In this paper, we present the derivation of Jeffreys divergence, generalized Fisher divergence, and the corresponding De Bruijn identities for space–time random field. First, we establish the connection between Jeffreys divergence and generalized Fisher information of a single space–time random field with respect to time and space variables. Furthermore, we obtain the Jeffreys divergence between two space–time random fields obtained by different parameters under the same Fokker–Planck equations. Then, the identities between the partial derivatives of the Jeffreys divergence with respect to space–time variables and the generalized Fisher divergence are found, also known as the De Bruijn identities. Later, at the end of the paper, we present three examples of the Fokker–Planck equations on space–time random fields, identify their density functions, and derive the Jeffreys divergence, generalized Fisher information, generalized Fisher divergence, and their corresponding De Bruijn identities.