Exact stationary state of a d-dimensional run-and-tumble particle in a harmonic potential.

We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in d dimensions confined in an isotropic harmonic trap V(r)=μr^{2}/2, with r=∥r∥. Rotational invariance reduces the problem to the stationary single-coordinate marginal p_{X}(x), from which the radial distribution p_{R}(r) and the full joint stationary density follow by explicit integral transforms. We first focus on a generalized trapped RTP in one dimension, where post-tumble velocities are drawn from an arbitrary distribution W(v). Using a Kesten-type recursion, we represent its stationary position in terms of a stick-breaking (or Dirichlet) process, yielding closed-form expressions for its distribution and its moments. Specializing W(v) to the projected velocity law of an isotropic RTP, we reconstruct p_{R}(r) and the full joint distribution of all the coordinates in d=1,2,3. In d=1 and d=2, the radial law simplifies to a beta distribution, while in d=3, we derive closed-form expressions for p_{R}(r) and the stationary joint distribution P(x,y,z), which differ from a beta distribution. In all cases, we characterize a persistence-controlled shape transition at the turning surface r=v_{0}/μ, where v_{0} is the self-propulsion speed. We further include thermal noise characterized by a diffusion coefficient D>0, showing that the stationary law is a Gaussian convolution of the D=0 result, which regularizes turning-point singularities and controls the crossover between persistence- and diffusion-dominated regimes as D→0 and D→∞, respectively. All analytical predictions are systematically validated against numerical simulations.