Wasserstein error estimates between telegraph processes and Brownian motion

2025/09/15 by Gerardo Barrera, Jani Lukkarinen, Barrera, Gerardo +3
Biochemistry, Genetics and Molecular Biology · #35L99 #60G50 #60J65 #60K50 #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Secondary 60G50

paper · pdf · doi:10.48550/arxiv.2509.11871

openalex publication_date 2025/09/15 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We provide non-asymptotic error bounds in the path Wasserstein distance with quadratic integral cost between suitable functionals of the telegraph process and the corresponding functional of Brownian motion with explicit diffusivity constant. These results cover, in particular, the well-known example of the exponential integral functional of the geometric Brownian motion. The non-asymptotic error bounds tend to zero in the so-called Kac regime. Moreover, the error bounds remain valid when the flip rate for the telegraph process is small. We assess the sharpness of the error bounds through numerical experiments.

Citations

• On the telegraph process driven by geometric counting process with Poisson-based resetting
• Multidimensional random motions with a natural number of finite velocities
• Telegraph random evolutions on a circle
• On the Exact Distributions of the Maximum of the Asymmetric Telegraph Process
• Telegraph process with elastic boundary at the origin
• On a directionally reinforced random walk
• Occupation time distributions for the telegraph process
• Estimation for the discretely observed telegraph process
• The equation of symmetric Markovian random evolution in a plane
• The distance between the Kac process and the Wiener process with applications to generalized telegraph equations

Related

• Wasserstein decay of one dimensional jump-diffusions
• Wasserstein Asymptotics for Brownian Motion on the Flat Torus and Brownian Interlacements
• Wasserstein distance estimates for jump-diffusion processes
• Evolution of the Wasserstein distance between the marginals of two Markov processes
• Exact rate of convergence for the empirical measure of a subordinated process in p-Wasserstein distance