  We study cosmological self-reproduction in models of inflation driven by a
scalar field $\phi$ with a noncanonical kinetic term ($k$-inflation). We
develop a general criterion for the existence of attractors and establish
conditions selecting a class of $k$-inflation models that admit a unique
attractor solution. We then consider quantum fluctuations on the attractor
background. We show that the correlation length of the fluctuations is of order
$c_{s}H^{-1}$, where $c_{s}$ is the speed of sound. By computing the magnitude
of field fluctuations, we determine the coefficients of Fokker-Planck equations
describing the probability distribution of the spatially averaged field $\phi$.
The field fluctuations are generally large in the inflationary attractor
regime; hence, eternal self-reproduction is a generic feature of $k$-inflation.
This is established more formally by demonstrating the existence of stationary
solutions of the relevant FP equations. We also show that there exists a
(model-dependent) range $\phi_{R}<\phi<\phi_{\max}$ within which large
fluctuations are likely to drive the field towards the upper boundary
$\phi=\phi_{\max}$, where the semiclassical consideration breaks down. An exit
from inflation into reheating without reaching $\phi_{\max}$ will occur almost
surely (with probability 1) only if the initial value of $\phi$ is below
$\phi_{R}$. In this way, strong self-reproduction effects constrain models of
$k$-inflation.
