I was trying to understand difference between drift and trend wherein I came across concepts of unit roots and trend stationary. (I haven't read any books on time series, just going through web).
This question came to my mind when I read this Wikipedia article on unit roots
It says, for a unit root process with drift "Any non-zero value of the noise term, occurring for only one period, will permanently affect the value of $y_t$". This statement looks clear from the equation of $y_t$
$y_t = y_{t-1} + c + e_t$
However, for a trend stationary process, it says "any transient noise will not alter the long-run tendency for $y_t$ to be on the trend line, as also shown in the graph." I can't figure out the use of long-run here. For example, taking the equation given in Wikipedia itself
$y_t = kt + u_t$
and considering $u_t$ to be normally distributed with 0 mean, any deviation will come back to trendline at the next time point itself. Will it not be true for all trend stationary seires.