  We calculate the mean shape of transition paths and first-passage paths based
on the one-dimensional Fokker-Planck equation in an arbitrary free energy
landscape including a general inhomogeneous diffusivity profile. The transition
path ensemble is the collection of all paths that do not revisit the start
position $x_A$ and that terminate when first reaching the final position $x_B$.
In contrast, a first-passage path can revisit but not cross its start position
$x_A$ before it terminates at $x_B$. Our theoretical framework employs the
forward and backward Fokker-Planck equations as well as first-passage, passage,
last-passage and transition-path time distributions, for which we derive the
defining integral equations. We show that the mean time at which the transition
path ensemble visits an intermediate position $x$ is equivalent to the mean
first-passage time of reaching the starting position $x_A$ from $x$ without
ever visiting $x_B$. The mean shape of first-passage paths is related to the
mean shape of transition paths by a constant time shift. Since for large
barrier height $U$ the mean first-passage time scales exponentially in $U$
while the mean transition path time scales linearly inversely in $U$, the time
shift between first-passage and transition path shapes is substantial. We
present explicit examples of transition path shapes for linear and harmonic
potentials and illustrate our findings by trajectories generated from Brownian
dynamics simulations.
