Epilogue to Volume IV: Bringing an Entire Motion into Mathematics

A microscope leaves records of positions; motion takes place between them. Writing down the joint distribution at every finite collection of times tells us how to compare those records. It does not by itself justify a continuous curve between them. A random spike can evade every observation time fixed in advance; increasingly narrow oscillations can hide between increasingly fine sampling points. To make an entire motion a mathematical object, we must control precisely what finite observations do not directly retain.

Continuity is therefore not a line added when drawing a picture. Once common increment estimates ensure a continuous modification, records on a countable dense set of times can determine the whole continuous path. We preserve the probabilistic information supplied by finite-dimensional distributions and prove that it can unfold consistently in a measurable path space. Path existence requires both parts.

Constructing a path and proving that a sequence of random paths approaches it remain different tasks. The latter also requires control over the whole family: increasingly violent motion must not escape into increasingly short gaps in time. Tightness turns this requirement into common conditions we can check. We first retain most of the probability on path sets subject to uniform control, then use finite observations to identify possible limits. Only then does the Gaussian limit of a random walk become a conclusion about an entire motion rather than a succession of position distributions.

Even “approaching” has more than one meaning. Moving the time of a jump slightly can leave a uniform distance as large as the whole jump height. A small adjustment of time can instead retain the same jump while disregarding a small timing error. This choice cannot erase arbitrary oscillations. A topology specifies which perturbations are small and determines which observations remain stable under passage to the limit. It is part of the limiting assertion.

Once the whole path comes into view, the distance between continuity and smoothness becomes apparent. A Brownian path has no velocity in the ordinary sense, yet its oscillation scales and quadratic variation can be identified rigorously. With jumps allowed, infinitely many small jumps may sometimes be summed directly; in other cases we must first subtract their mean contribution and control the fluctuations left over. Finite-time distribution calculations remain useful, but they must work together with these path structures.

Another route seems to run in the opposite direction: rather than follow one motion, average a function of its future position. Random transitions become a deterministic evolution of operators, and their short-time changes give a generator. Yet the same interior differential formula can accommodate different boundary behavior. Only after specifying the domain, the semigroup conditions, and the requirements for path realization can a local formula determine a random motion in the reverse direction. Averaging has not removed randomness; it has expressed its consistently transmitted structure as an equation.

Exceptionally rare paths require a further change of scale. Ordinary weak convergence describes the overall limit of path distributions, but that limit alone cannot distinguish how small vanishing probabilities are. Large deviations compare path neighborhoods whose probabilities vanish and identify their different exponential costs. For small-noise Brownian motion, squared energy measures the cost of a deviation route. It does not mean that typical Brownian paths suddenly acquire an integrable velocity. Ordinary and exponential tightness, weak approximation and control of errors on an exponential scale, answer different questions.

An entire motion can now enter probability theory: we can establish its existence, compare its shape, study its averages, and measure the cost of its departures from typical behavior. But accumulating a quantity along such a rough path still cannot simply follow the rules for smooth curves. What information may be used as we accumulate, and what will the small fluctuations and jumps leave behind? These questions bring path geometry together with information evolving in time. After understanding the shape of a motion, we still need to learn how to calculate along it.