= Sample path
{title2=$t\mapsto X_t(\omega)$}
= Sample trajectory
{synonym}
For a <stochastic process> $X_t$ and a fixed <sample space> outcome $\omega$, its sample path is the <function> $t\mapsto X_t(\omega)$. One row of observed fair bits, one trajectory of a <random walk>, or one staircase of <Poisson process> arrival counts is a <sample path>. Fixing $t$ instead gives a <random variable> ranging across possible histories. The same two-variable object $X_t(\omega)$ therefore describes both observations across outcomes and evolution within an outcome.
Statements about <continuous>, <càdlàg> or <Hölder continuous function> <sample paths> concern these <functions> simultaneously in time. They differ from assertions made at each fixed time separately. For example, take $U$ with a <uniform distribution> on $[0,1]$ and set $X_t=\mathbf1_{\{t=U\}}$. For every fixed $t$, $X_t=0$ <almost surely>, but every <sample path> has a single spike. Thus this process is a <modification of a stochastic process> that is identically zero, while the two are not <indistinguishable stochastic processes>. <Continuity> of both sets of <sample paths> rules out this discrepancy: equality at a countable dense set then implies equality everywhere. This is why path regularity is an additional part of constructing a continuous-time model.
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