  This paper studies the capacity of molecular communications in fluid media,
where the information is encoded in the number of transmitted molecules in a
time-slot (amplitude shift keying). The propagation of molecules is governed by
random Brownian motion and the communication is in general subject to
inter-symbol interference (ISI). We first consider the case where ISI is
negligible and analyze the capacity and the capacity per unit cost of the
resulting discrete memoryless molecular channel and the effect of possible
practical constraints, such as limitations on peak and/or average number of
transmitted molecules per transmission. In the case with a constrained peak
molecular emission, we show that as the time-slot duration increases, the input
distribution achieving the capacity per channel use transitions from binary
inputs to a discrete uniform distribution. In this paper, we also analyze the
impact of ISI. Crucially, we account for the correlation that ISI induces
between channel output symbols. We derive an upper bound and two lower bounds
on the capacity in this setting. Using the input distribution obtained by an
extended Blahut-Arimoto algorithm, we maximize the lower bounds. Our results
show that, over a wide range of parameter values, the bounds are close.
