Biased random walks with finite mean first passage time

A power-law distance-dependent biased random walk model with a tuning parameter ($\sigma$) is introduced in which finite mean first passage times are realizable if $\sigma$ is less than a critical value $\sigma_c$. We perform numerical simulations in $1$-dimension to obtain $\sigma_c \sim 1.14$. The three-dimensional version of this model is related to the phenomenon of chemotaxis. Diffusiophoretic theory supplemented with coarse-grained simulations establish the connection with the specific value of $\sigma = 2$ as a consequence of in-built solvent diffusion. A variant of the one-dimensional power-law model is found to be applicable in the context of a stock investor devising a strategy for extricating their portfolio out of loss.