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.