Risk Assessment of Noise-Limit Exceedance Using Banach Random Walk in Noisy Space

Abstract

We develop a probabilistic method for assessing the risk that the level generated by a noise source exceeds a permissible limit. The physical input is a sequence of acoustic pressure pulses represented by their energy contributions, while the regulatory output is a decibel level compared with that limit. The model uses the Banach random walk in noisy space (BRWNS), in which a source is steadily supplied with energy and randomly emits pulses into the environment. The exceedance risk is defined as the probability that at least one partial mean energy level in an observation window exceeds the permissible limit. We derive an explicit integral expression for this probability by transforming the decibel threshold back to an energy threshold. The result shows that the permissible limit determines the part of the stochastic energy domain over which the BRWNS density must be integrated. The interpretation is linked to classical acoustic level operations, including energetic addition of levels and background-noise correction. A road-traffic example illustrates how environmental limits can be translated into the model threshold. Monte Carlo simulations for the standard BRWNS show a sharp increase in the exceedance probability when the source energy becomes larger than the transformed permissible limit.