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We consider two particles performing continuous-time nearest neighbor randomwalk on $\mathbb Z$ and interacting with each other when they are atneighboring positions. Typical examples are two particles in the partialexclusion process or in the inclusion process. We provide an exact formula forthe Laplace-Fourier transform of the transition probabilities of thetwo-particle dynamics. From this we derive a general scaling limit result,which shows that the possible scaling limits are coalescing Brownian motions,reflected Brownian motions, and sticky Brownian motions. In particle systemswith duality, the solution of the dynamics of two dual particles providesrelevant information. We apply the exact formula to the the symmetric inclusionprocess, that is self-dual, in the condensation regime. We thus obtain tworesults. First, by computing the time-dependent covariance of the particleoccupation number at two lattice sites we characterize the time-dependentcoarsening in infinite volume when the process is started from a homogeneousproduct measure. Second, we identify the limiting variance of the density fieldin the diffusive scaling limit, relating it to the local time of stickyBrownian motion.

Exact formulas for two interacting particles and applications in particle systems with duality