It is common in the theory of stochastic processes to "build" stochastic processes by imposing properties on the distribution of the process (see Brownian motion for obvious example: independent normal increments...) and then deduce properties of the sample paths of the process (for the case of Brownian motion, nowhere differentiability $a.s$).
What if we want to do it the other way ? We impose strong sample paths properties, and then ask what are the distributional properties of such a process ?
Here is an example.
Suppose we would like to learn about the distribution of a stochastic process $X_t$ on $(0,1)$ such that the sample paths of the process are $a.s$ continuous, non-decreasing, $X(0^+) = -\infty$, $X(1^-)=\infty$, $\int_0^1 X_t^2 \,dt < \infty$.
What are the distributional properties of such a process ? What are the tools at our disposal to study such a problem ?