  Locally activated random walks are defined as random processes, whose
dynamical parameters are modified upon visits to given activation sites. Such
dynamics naturally emerge in living systems as varied as immune and cancer
cells interacting with spatial heterogeneities in tissues, or at larger scales
animals encountering local resources. At the theoretical level, these random
walks provide an explicit construction of strongly non Markovian, and aging
dynamics. We propose a general analytical framework to determine various
statistical properties characterizing the position and dynamical parameters of
the random walker on $d$-dimensional lattices. Our analysis applies in
particular to both passive (diffusive) and active (run and tumble) dynamics,
and quantifies the aging dynamics and potential trapping of the random walker;
it finally identifies clear signatures of activated dynamics for potential use
in experimental data.
