  Gravitational-wave memory effects are identified by their distinctive effects
on families of freely falling observers: after a burst of waves pass by their
locations, memory effects can cause lasting relative displacements of the
observers. These effects are closely related to the infrared properties of
gravity and other massless field theories, including their asymptotic
symmetries and conserved quantities. In this paper, we investigate the
connection between memory effects, symmetries, and conserved quantities in
Brans-Dicke theory. We compute the field equations in Bondi coordinates, and we
define a set of boundary conditions that represent asymptotically flat
solutions in this context. Next, we derive the asymptotic symmetry group of
these spacetimes, and we find that it is the same as the Bondi-Metzner-Sachs
group in general relativity. Because there is an additional polarization of
gravitational waves in Brans-Dicke theory, we compute the memory effects
associated with this extra polarization (the so-called "breathing" mode). This
breathing mode produces a uniform expansion (or contraction) of a ring of
freely falling observers. After these breathing gravitational waves pass by the
observers' locations, there are two additional memory effects that depend on
their initial displacements and relative velocities. Neither of these
additional memory effects seems to be related to asymptotic symmetries or
conserved quantities; rather, they are determined by the properties of the
nonradiative region before and after the bursts of the scalar field and the
gravitational waves. We discuss the properties of these regions necessary to
support nontrivial breathing-mode-type memory effects.
