  Observational data suggest that a belt of planetesimals is expected close to
the snow line in protoplanetary disks. Assuming there is such a belt in
TRAPPIST-1 system, we examine possibilities of water delivery to the planets
via planetesimals from the belt. The study is accomplished by numerical
simulations of dynamical evolution of a hypothetical planetesimal belt. Our
results show that the inner part of the belt is dynamically unstable and
planetesimals located in this region are quickly scattered away, with many of
them entering the region around the planets. The main dynamical mechanism
responsible for the instability are close encounters with the outermost planet
Trappist-1h. A low-order mean-motion resonance 2:3 with Trappist-1h, located in
the same region, also contributes to the objects transport. In our nominal
model, the planets have received non-negligible amount of water, with the
smallest amount of $15$\% of the current Earth's water amount (EWA) being
delivered to the planet 1b, while the planets Trappist-1e and Trappist-1g have
received more than $60$\% of the EWA. We have found that while the estimated
efficiency of water transport to the planets is robust, the amount of water
delivered to each planet may vary significantly depending on the initial masses
and orbits of the planets. The estimated dynamical "half-lives" have shown that
the impactors' source region should be emptied in less then 1~Myr. Therefore,
the obtained results suggest that transport of planetesimals through the system
preferably occurs during an early phase of the planetary system evolution.
