2025/06/27 by Kyprianou, Andreas E., Mantelos, Harry S., Rivero, Victor
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2506.22020
We start by remarking a one-to-one correspondence between self-similar Markov processes (ssMps) on a Banach space and Markov additive processes (MAPs) that is analogous to the well-known one between positive ssMps and Lévy processes through the renowned Lamperti-transform, with the main difference that ours is norm-dependent. We then consider multidimensional self-similar Markov processes obtained by killing or by reflecting a stable process or Brownian motion in the orthant and we then fully describe the MAPs associated to them using the L1-norm. Namely, we describe the MAP underlying the ssMp obtained by killing a d-dimensional α-stable process when it leaves the orthant and the one obtained by reflecting it back in the orthant continuously (or by a jump); finally, we also describe the MAP underlying d-dimensional Brownian motion reflected in the orthant. The first three of the aforementioned examples are pure-jump, and the last is a diffusion, so their characterization is given through their Lévy system, generator and/or through the modulated SDE that defines them, respectively.