Decide which of these are standard Brownian motions and justify each: X_t = (1/c) W_{c^2 t} for c > 0; Y_t = t W_{1/t} for t > 0 with Y_0 = 0; Z_t = W_T - W_{T-t} on [0,T].

Decide which of these are standard Brownian motions and justify each: X_t = (1/c) W_{c^2 t} for c > 0; Y_t = t W_{1/t} for t > 0 with Y_0 = 0; Z_t = W_T - W_{T-t} on [0,T].

Approach: A centred Gaussian process with continuous paths and covariance min(s,t) is standard Brownian motion. Compute each covariance, then treat continuity at the origin separately for the inverted process.

All three are standard Brownian motions. Each is a centred Gaussian process, so only the covariance function and path continuity need checking. Scaling invariance: Cov(X_s, X_t) = (1/c^2) min(c^2 s, c^2 t) = min(s,t). Time inversion: Cov(Y_s, Y_t) = st * min(1/s, 1/t) = st / max(s,t) = min(s,t), and continuity at 0 is the only real work, since it is equivalent to W_u / u tending to 0 as u grows, which follows from the law of large numbers applied along integers plus the maximal inequality between them. Time reversal: Cov(Z_s, Z_t) = Cov(W_T - W_{T-s}, W_T - W_{T-t}) = T - (T-s) - (T-t) + min(T-s, T-t) = min(s,t) for s,t in [0,T]. Time inversion is the useful one, because it converts long horizon statements about W into small time statements at the origin.

Follow-up: Under time inversion, what does the law of the iterated logarithm at infinity become as a statement about the path near time zero?

Key concepts: scaling invariance, time inversion, covariance function, time reversal.