Analytical expression for the optimal curvature in informed search with parabolic resetting

Determine an analytical expression for the optimal curvature parameter r0/b^2 that minimizes the mean first-passage time for a one-dimensional diffusive searcher starting at distance L from a target at the origin under the position-dependent resetting rate r(x) ≈ (r0/b^2) x^2 in the regime b ≫ L, where b is the environmental sensing length scale.

Background

The paper studies a one-dimensional diffusive searcher that resets to its initial position with a position-dependent rate r(x) capturing environmental information through a sigmoidal form r(x)=r0(1−1/(1+(x/b)^2)). In the regime b ≫ L, this rate is well approximated by a parabolic form r(x) ≈ (r0/b^2) x^2, introducing a single curvature parameter r0/b^2.

The authors argue that the mean first-passage time (MFPT) diverges when the curvature is either too small or too large, implying the existence of an optimal curvature. They numerically estimate this optimal value but explicitly note the absence of an analytical expression, making its derivation an open question relevant to optimizing informed search strategies.