Superdiffusive planar random walks with polynomial space-time drifts

Conrado da Costa, Mikhail Menshikov, Vadim Shcherbakov, and Andrew Wade

Stochastic Processes and their Applications, 176, October 2024, article 104420. DOI: 10.1016/j.spa.2024.104420

Supported by EPSRC award Anomalous diffusion via self-interaction and reflection (EP/W00657X/1).

Abstract

We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent 3/4. The self-interacting process originated in discussions with Francis Comets.