Tractable Infinite‐Dimensional Model for Long‐Term Environmental Impact Assessment of Long‐Memory Decay Processes

ABSTRACT Focusing on the assessment of benthic algae blooms that decay subexponentially, we propose a tractable (solvable in a semi‐explicit 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 decay 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 solve this system, verify a reasonable solution, and numerically handle it 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 laboratory experiment. This paper offers a tractable framework for the assessment of persistent environmental phenomena.