We derive the exact nonequilibrium steady state of a run-and-tumble particle (RTP) in $d$ dimensions confined in an isotropic harmonic trap $V\left(\mathbf{r}\right)=\mu r^2/2$, with $r=\parallel \mathbf{r}\parallel$. Rotational invariance reduces the problem to the stationary single-coordinate marginal $p_X\left(x\right)$, from which the radial distribution $p_R\left(r\right)$ 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 $W\left(v\right)$. 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 $W\left(v\right)$ to the projected velocity law of an isotropic RTP, we reconstruct $p_R\left(r\right)$ 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 $p_R\left(r\right)$ 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 $r=v_0/\mu$, 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 $D\to 0$ and $D\to \infty$, respectively. All analytical predictions are systematically validated against numerical simulations.

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• Brownian motion
• Nonequilibrium statistical mechanics
• Random walks
• Stochastic processes