Aging dynamics of $d$−dimensional locally activated random walks

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.