Tractable infinite-dimensional model for long-term environmental impact assessment of long-memory processes

TL;DR

This paper introduces a mathematically tractable model for assessing long-term environmental impacts of long-memory processes, specifically applied to benthic algae dynamics, using a novel infinite-dimensional approach and numerical solutions.

Contribution

It develops a closed-form, well-defined environmental index for long-memory processes, solved via an infinite-dimensional Hamilton-Jacobi-Bellman system, with practical application to algae population dynamics.

Findings

Successfully models long-memory environmental impacts.

Provides a numerical method for solving complex infinite-dimensional systems.

Demonstrates applicability through a river algae case study.

Abstract

Focusing on the assessment of benthic algae blooms that decay subexponentially, we propose a tractable (solvable in a closed form) and well-defined (that does not diverge) environmental index for the impact assessment of long-memory processes under model uncertainties. Our target system generates long memory through an infinite superposition of multiscale processes. The sensitivity of the environmental index can be controlled by the degree of model uncertainty in terms of the relative entropy and nonexponential discount; hence, we apply a long-memory discount to evaluate long-memory processes. In our framework, the evaluation of the environmental index is reduced to finding a proper solution to an infinite-dimensional extended Hamilton-Jacobi-Bellman system. We can solve this system under sufficient conditions for the unique existence of sufficiently regular solutions, and numerically handle them by using a quantization technique. Finally, we present a demonstrative application of the proposed framework to benthic algae population dynamics in river environments based on a laboratorial experiment. This paper offers a tractable framework towards the assessment of persistent environmental phenomena.