Differential Properties of Information in Jump-diffusion Channels

TL;DR

This work addresses how entropy and mutual information evolve in continuous-time jump-diffusion channels by modeling the output as a Markov process with drift, diffusion, and jumps.It develops two complementary analytical tools: a Kramers–Moyal–based series expansion and a Kolmogorov–Feller–based integral form, expressing the time derivatives in terms of Fisher-type information and mismatched KL divergences.A key contribution is the extension of de Bruijn’s identity and the I-MMSE relation to general jump-diffusion channels, including a precise decomposition of the mutual information derivative into diffusion and jump components, with additive-noise channels treated as a special case.The results provide a pathway to quantify information loss over time in complex channels and offer practical estimation advantages via propagator moments, with implications for advanced modeling in communications and related fields.

Abstract

We propose a channel modeling using jump-diffusion processes, and study the differential properties of entropy and mutual information. By utilizing the Kramers-Moyal and Kolmogorov-Feller equations, we express the mutual information between the input and the output in series and integral forms, presented by Fisher-type information and mismatched KL divergence. We extend de Bruijn's identity and the I-MMSE relation to encompass general Markov processes.

Theorems & Definitions (29)

• Lemma 1
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• Theorem 1
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• Theorem 2
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