Potential Theory and Hitting Probabilities

Stochastic processes rooted in potential theory provide a rigorous mathematical language for quantifying healthcare access, measuring equity, and assessing systemic risk in complex environments. This chapter develops hitting probabilities and first-passage time theory within adaptive healthcare systems. Beginning with Brownian motion and harmonic functions, it advances through diffusion models, Markov decision processes, epidemic random walks, and Lévy jump-diffusion frameworks. Each construct is linked to healthcare scenarios including patient trajectory modelling, epidemic boundary crossing, treatment optimisation, and resource allocation under uncertainty. Attention is given to how probabilistic boundaries encode structural inequities between rural and urban populations. Mathematical derivations are supported by figures, a graph, and a comparative table to provide a unified stochastic toolkit for planning.

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