Kolmogorov’s Continuity Criterion
Kolmogorov sought the condition that a stochastic process must satisfy in order to have a continuous modification. The key lies in a good bound for when is close to ; by Markov’s inequality this depends on the moments of .
A second question: can we measure the “degree of smoothness” of the (continuous) sample paths thus obtained?
Lévy’s Modulus of Continuity for Brownian Motion
Let be a standard Brownian motion.
Scaling and Fractal Properties of Brownian Motion
Many natural sets derived from Brownian sample paths can be regarded as “random fractals”. This relies on the scaling invariance property.
Proof. Path continuity and the independence and stationarity of increments are preserved because ◻
Let and define the first exit time Using and setting , we have corresponds to , and corresponds to . Hence In particular, , i.e., a constant multiple of . Moreover, which is a function of the ratio only.
Time Inversion
Define the process by Then is also a standard Brownian motion.
Proof. The finite‑dimensional distributions are Gaussian with mean zero and covariance . For , so the f.d.d. agree with those of . Continuity for is clear. For , note that the distribution of is the same as that of a Brownian motion; hence almost surely. Because is dense in and the paths are continuous on almost surely, the process has continuous sample paths almost surely. ◻
Consequences of Scaling and Inversion
The scaling and inversion invariances tie Brownian motion to two important groups of transformations on . These symmetries are extremely useful.
Scaling invariance: if we have one interval of a second of a Brownian path, we can expand it to an interval of seconds of an equally valid Brownian path.
Inversion invariance: the first second of the life of a Brownian path is rich enough to capture the behavior of a Brownian path from the end of the first second until the end of time.