  We study the dynamics of Brownian particles in a heterogeneous
one-dimensional medium with a spatially-dependent diffusion coefficient of the
form $D(x)\sim |x|^c$, at constant temperature. The particle's probability
distribution function (PDF) is calculated both analytically, by solving Fick's
diffusion equation, and from numerical simulations of the underdamped Langevin
equation. At large times, the PDFs calculated by both approaches yield
identical results, corresponding to subdiffusion for $c<0$, and superdiffusion
for $0<c<1$. For $c>1$, the diffusion equation predicts that the particles
accelerate. Here, we show that this phenomenon, previously considered in
several works as an illustration for the possible dramatic effects of
spatially-dependent thermal noise, is unphysical. We argue that in an
isothermal medium, the motion cannot exceed the ballistic limit ($\left\langle
x^2\right\rangle \sim t^2$). The ballistic limit is reached when the friction
coefficient drops sufficiently fast at large distances from the origin, and is
correctly captured by Langevin's equation.
