Abridged abstract: Inert interactions between randomly moving entities and
spatial disorder play a crucial role in quantifying the diffusive properties of
a system. These interactions affect only the movement of the entities, and
examples range from molecules advancing along dendritic spines to anti-predator
displacements of animals due to sparse vegetation. Despite the prevalence of
such systems, a general framework to model the movement explicitly in the
presence of spatial heterogeneities is missing. Here, we tackle this challenge
and develop an analytic theory to model inert particle-environment interactions
in domains of arbitrary shape and dimensions. We use a discrete space
formulation which allows us to model the interactions between an agent and the
environment as perturbed dynamics between lattice sites. Interactions from
spatial disorder, such as impenetrable and permeable obstacles or regions of
increased or decreased diffusivity, as well as many others, can be modelled
using our framework. We provide exact expressions for the generating function
of the occupation probability of the diffusing particle and related transport
quantities such as first-passage, return and exit probabilities and their
respective means. We uncover a surprising property, the disorder indifference
phenomenon of the mean first-passage time in the presence of a permeable
barrier in quasi-1D systems. We demonstrate the widespread applicability of our
formalism by considering three examples that span across scales and
disciplines. (1) We explore an enhancement strategy of transdermal drug
delivery. (2) We associate the disorder with a decision-making process of an
animal to study thigmotaxis. (3) We illustrate the use of spatial
heterogeneities to model inert interactions between particles by modelling the
search for a promoter region on the DNA by transcription factors during gene
transcription