Optimal mean first-passage time of a run-and-tumble particle in a class of one-dimensional confining potentials

We consider a run-and-tumble particle (RTP) in one dimension, subjected to a telegraphic noise with a constant rate γ , and in the presence of an external confining potential <?CDATA V(x) = α |x|ᵖ ?> with <?CDATA p ≥1 ?> . We compute the mean first-passage time (MFPT) at the origin <?CDATA τ _γ (x₀) ?> for an RTP starting at x 0 . We obtain a closed form expression for <?CDATA τ _γ (x₀) ?> for all <?CDATA p ≥1 ?> , which becomes fully explicit in the case <?CDATA $p=1, p=2$ ?> and in the limit <?CDATA p → ∞ ?> . For generic <?CDATA $p>1$ ?> we find that there exists an optimal rate <?CDATA γ ₒₚₜ ?> that minimizes the MFPT and we characterize in detail its dependence on x 0 . We find that <?CDATA γ ₒₚₜ ∝1/x₀ ?> as <?CDATA x₀\,→0 ?> , while <?CDATA γ ₒₚₜ ?> converges to a non-trivial constant as <?CDATA x₀\,→ ∞ ?> . In contrast, for p = 1, there is no finite optimum and <?CDATA γ ₒₚₜ → ∞ ?> in this case. These analytical results are confirmed by our numerical simulations.