  We derive here Lagrangian fluctuation-dissipation relations for advected
scalars in wall-bounded flows. The relations equate the dissipation rate for
either passive or active scalars to the variance of scalar inputs from the
initial values, boundary values, and internal sources, as those are sampled
backward in time by stochastic Lagrangian trajectories. New probabilistic
concepts are required to represent scalar boundary conditions at the walls: the
boundary local-time density at points on the wall where scalar fluxes are
imposed and the boundary first hitting-time at points where scalar values are
imposed. These concepts are illustrated both by analytical results for the
problem of pure heat conduction and by numerical results from a database of
channel-flow flow turbulence, which also demonstrate the scalar mixing
properties of near-wall turbulence. As an application of the
fluctuation-dissipation relation, we examine for wall-bounded flows the
relation between anomalous scalar dissipation and Lagrangian spontaneous
stochasticity, i.e. the persistent non-determinism of Lagrangian particle
trajectories in the limit of vanishing viscosity and diffusivity. In the first
paper of this series, we showed that spontaneous stochasticity is the only
possible mechanism for anomalous dissipation of passive or active scalars, away
from walls. Here it is shown that this remains true when there are no scalar
fluxes through walls. Simple examples show, on the other hand, that a distinct
mechanism of non-vanishing scalar dissipation can be thin scalar boundary
layers near the walls. Nevertheless, we prove for general wall-bounded flows
that spontaneous stochasticity is another possible mechanism of anomalous
scalar dissipation.
