Exact position distribution of a harmonically confined run-and-tumble particle in two dimensions

We consider an overdamped run-and-tumble particle in two dimensions, with self-propulsion in an orientation that stochastically rotates by 90^{∘} at a constant rate, clockwise or counterclockwise with equal probabilities. In addition, the particle is confined by an external harmonic potential of stiffness μ, and possibly diffuses. We find the exact time-dependent distribution P(x,y,t) of the particle's position, and in particular, the steady-state distribution P_{st}(x,y) that is reached in the long-time limit. We also find P(x,y,t) for a "free" particle, μ=0. We achieve this by showing that, under a proper change of coordinates, the problem decomposes into two statistically independent one-dimensional problems, whose exact solution has recently been obtained. We then extend these results in several directions, to two such run-and-tumble particles with a harmonic interaction, to analogous systems of dimension three or higher, and by allowing stochastic resetting.