In this presentation we will obtain a non-asymptotic process level control between the telegraph process (a.k.a. Goldstein--Kac equation/process) and a Brownian motion with explicit diffusivity constant via a transportation Wasserstein path-distance with quadratic average cost. We stress that the marginals of the telegraph process solves a partial linear differential equation of the hyperbolic type for which explicit computations can be carried out in terms of Bessel functions. In the present talk, I will discuss a probabilistic coupling approach, which is a robust technique that in principle can be used for more general PDEs. The proof is done via the interplay of the following probabilistic couplings: coin-flip coupling, synchronous coupling and the celebrated Komlós--Major--Tusnády coupling. Using the previous result, we derive a probabilistic coupling between a multivariate (non-commutative) geometric Brownian motion and the celebrated velocity flip model with quadratic interaction.

The talk is based on joint work with Jani Lukkarinen, University of Helsinki, Finland.