  We consider the trapping reaction, $A+B\to B$, where $A$ and $B$ particles
have a diffusive dynamics characterized by diffusion constants $D_A$ and $D_B$.
The interaction with $B$ particles can be formally incorporated in an effective
dynamics for one $A$ particle as was recently shown by Bray {\it et al}. [Phys.
Rev. E {\bf 67}, 060102 (2003)]. We use this method to compute, in space
dimension $d=1$, the asymptotic behaviour of the spatial fluctuation,
$<z^2(t)>^{1/2}$, for a surviving $A$ particle in the perturbative regime,
$D_A/D_B\ll 1$, for the case of an initially uniform distribution of $B$
particles. We show that, for $t\gg 1$, $<z^2(t)>^{1/2} \propto t^{\phi}$ with
$\phi=1/4$. By contrast, the fluctuations of paths constrained to return to
their starting point at time $t$ grow with the larger exponent 1/3. Numerical
tests are consistent with these predictions.
