The sojourn time problem for a $p$-adic random walk and its applications to the spectral diffusion of proteins

We consider the problem of the distribution of the sojourn time in a compact set $\mathbb{Z}_{p}$ in the case of a $p$-adic random walk. We rely on the results of our previous studies of the distribution of the first return time for a $p$-adic random walk and the results of Takacs on the study of the sojourn time problem for a wide class of random processes. For a $p$-adic random walk we find the mean sojourn time of the trajectory in $\mathbb{Z}_{p}$ and the asymptotics as $t\rightarrow\infty$ of arbitrary moments of the distribution of the sojourn time in $\mathbb{Z}_{p}$. We also discuss some possible applications of our results to the modeling of relaxation processes related to the conformational dynamics of protein.