First passage of a diffusing particle under stochastic resetting in bounded domains with spherical symmetry

We investigate the first passage properties of a Brownian particle diffusing freely inside a $d$-dimensional sphere with absorbing spherical surface subject to stochastic resetting. We derive the mean time to absorption (MTA) as functions of resetting rate $\gamma$ and initial distance $r$ of the particle to the center of the sphere. We find that when $r>r_c$ there exists a nonzero optimal resetting rate $\gamma_{{\rm opt}}$ at which the MTA is a minimum, where $r_c=\sqrt {d/\left( {d + 4} \right)} R$ and $R$ is the radius of sphere. As $r$ increases, $\gamma_{{\rm opt}}$ exhibits a continuous transition from zero to nonzero at $r=r_c$. Furthermore, we consider that the particle lies in between two two-dimensional or three-dimensional concentric spheres, and obtain the domain in which resetting expedites the MTA, which is $(R_1, r_{c_1}) \cup (r_{c_2},R_2)$, with $R_1$ and $R_2$ being the radius of inner and outer spheres, respectively. Interestingly, when $R_1/R_2$ is less than a critical value, $\gamma_{{\rm opt}}$ exhibits a discontinuous transition at $r=r_{c_1}$; otherwise, such a transition is continuous. However, at $r=r_{c_2}$, $\gamma_{{\rm opt}}$ always shows a continuous transition.