Introduction

The goal of these notes is to answer two fundamental questions about stochastic processes:

1. Given a process defined by its finite‑dimensional distributions, when can we guarantee that it has a continuous version (i.e., a modification with continuous sample paths)?

2. If a continuous version exists, how smooth are its sample paths? Can we measure their irregularity quantitatively?

The answers are provided by two landmark results:

• The Kolmogorov continuity theorem (also called Kolmogorov–Čentsov theorem) gives a simple moment condition that implies the existence of a Hölder‑continuous modification.

• Lévy’s modulus of continuity for Brownian motion tells us exactly how fast the increments can oscillate: the almost sure oscillation is of order , not as one might naively think.

We will also explore the scaling and inversion symmetries of Brownian motion, which are the source of its self‑similarity and fractal nature. The problems at the end illustrate these concepts and lead to deeper insights.

Hölder Continuity – A Measure of Smoothness

Before stating the theorems, we need a way to grade the regularity of a continuous function.

Why is this important? For stochastic processes, sample paths are almost always irregular (e.g., Brownian motion is nowhere differentiable). Yet they often satisfy a Hölder condition with some exponent . The Kolmogorov theorem tells us that if the moments of the increments grow like a power of the time difference, then the paths are automatically Hölder continuous – we don’t need to construct them explicitly.

Kolmogorov’s Continuity Theorem

Intuition

The condition controls the -th moment of the increment. By Markov’s inequality, If we take with , the right‑hand side becomes . For small , this probability decays faster than , which allows us to control the oscillations on a fine dyadic grid and then use a Borel–Cantelli argument. The result is a continuous, Hölder‑regular modification.

Example: Brownian motion

For standard Brownian motion in , we have . Hence where for . Take large and set . For we have . Then the theorem guarantees Hölder continuity for any . Letting , we obtain Hölder continuity for all . So Brownian paths are -Hölder for every , but not for (this is the content of Lévy’s modulus).

Lévy’s Modulus of Continuity for Brownian Motion

For Brownian motion, the optimal Hölder exponent is in a very precise, almost‑sure sense.

Interpretation

For each fixed , the maximum increment over a sliding window of length is approximately . This is slightly larger than because grows slowly as . The theorem says:

• If , then eventually (for all sufficiently small ),

• If , then infinitely often (for a sequence ),

Thus the function is the exact modulus of continuity for Brownian motion.

Consequences

1. Brownian motion is nowhere locally -Hölder for any (because the oscillation is too large). For it fails as well (the additional factor diverges). Hence the critical exponent is exactly .

2. The result is a cornerstone of stochastic calculus and the theory of Gaussian processes. It shows that Brownian paths are just barely continuous – but with an infinite -variation (quadratic variation is non‑zero).

Scaling and Inversion Symmetries of Brownian Motion

Brownian motion enjoys two important invariance properties that make it a “random fractal”. They are often used to deduce properties of first passage times, local times, and fractal dimensions.

Scaling invariance

Meaning: If you zoom out horizontally by and vertically by , the path looks like a Brownian motion again. This self‑similarity is the key to many fractal properties.

First exit times

Let , where . Using scaling, we can relate expectations and probabilities.

Time inversion

Significance: The behaviour of BM near (infinitesimal times) and near (large times) are the same after inversion. This is why properties like “almost every path hits every real number” can be derived from the fact that a.s.

Supplementary Examples and Applications

Example: Using scaling to compute exit probabilities

Let with . By scaling, Thus the probability depends only on the ratio . In fact, for BM, by symmetry, and one can show for . This matches the gambler’s ruin formula.

Example: Hölder regularity of BM via Kolmogorov

Take . Then . Here , so we have exponent (so ). Hence . This already gives Hölder of order . By taking larger we approach . So the theorem gives existence of a version that is -Hölder? No, it gives for any but not (the theorem would require which is impossible). Lévy’s modulus shows that fails.

Example: The law of the iterated logarithm (LIL)

A related result is Khinchin’s LIL: This is a pointwise version (at a fixed ) whereas Lévy’s modulus is uniform over the whole interval.

Problems with Detailed Solutions/Hints

Below are the problems from the original notes, expanded with full solutions or extensive hints.

Further Reading

• Kolmogorov’s original paper: “Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung” (1931).

• For Brownian motion: Karatzas & Shreve, Brownian Motion and Stochastic Calculus.

• For Gaussian processes and Hölder regularity: Adler & Taylor, Random Fields and Geometry.