Kolmogorov continuity theorem

The Kolmogorov continuity theorem provides conditions under which a stochastic process (a collection of random variables evolving over time) has a version with continuous paths. In simple terms, if the process's increments are controlled so that their probabilities decay quickly as the time difference shrinks, then there exists a way to modify the process slightly to ensure it behaves smoothly without jumps, making its paths continuous. This theorem is fundamental in probability theory for understanding when complex random systems can be interpreted as continuous functions, enabling further analysis and modeling of phenomena like noise or fluid flows.