Stochastic nonlocal traffic flow models with Markovian noise

TL;DR

This paper extends stochastic nonlocal traffic flow models by incorporating Markovian noise, analyzing stability, and introducing a local solution operator to understand the noise's impact through simulations.

Contribution

It introduces a Markovian noise model for nonlocal traffic flow, ensuring interpretability and boundedness, and develops a local solution operator for analyzing noise effects.

Findings

Markovian noise preserves boundedness and interpretability.

The local solution operator reveals the noise's local effects.

Simulations demonstrate the impact of different noise processes on traffic flow.

Abstract

We extend our recently introduced stochastic nonlocal traffic flow model to more general random perturbations, including Markovian noise derived from a discretized Jacobi-type stochastic differential equation. Invoking a deterministic stability estimate, we show that the arising random weak entropy solutions are measurable, ensuring that quantities such as the expectation are well-defined. We show that the proposed Jacobi-type noise is of particular interest as it ensures interpretability, preserves boundedness, and significantly alters the stochastic realizations compared to the previous white noise approach. Moreover, we introduce a local solution operator which provides information on the local effect of the noise and utilize it to derive a mean-value hyperbolic nonlocal PDE, which serves as a proxy for the mean value of the exact solution. The quality of this proxy and the impact of the noise process are analyzed in several simulation studies.