Random search with resetting in heterogeneous environments

We investigate random searches under stochastic position resetting at rate r, in a bounded 1D environment with space-dependent diffusivity D(x). For arbitrary shapes of D(x) and prescriptions of the associated multiplicative stochastic process, we obtain analytical expressions for the average time T for reaching the target (mean first-passage time), given the initial and reset positions, in good agreement with stochastic simulations. For arbitrary D(x), we obtain an exact closed-form expression for T, within a Stratonovich scenario, while for other prescriptions, like Itô and anti-Itô, we derive asymptotic approximations for small and large rates r. Exact results are also obtained for particular forms of D(x), such as the linear one, with arbitrary prescriptions, allowing to outline and discuss the main effects introduced by diffusive heterogeneity on a random search with resetting. We explore how the effectiveness of resetting varies with different types of heterogeneity.