Lawler defined BM with 2 params – drift and variance v, but the meaning of variance is tricky.
Note a BM is about a TVRV and notice the difference between a N@T vs TVRV. A N@T could be modeled by a Gaussian variable with a variance. The variance v of a BM is about the variance of increment. Specifically, the increment over deltaT is a regular Gaussian RV with a variance = deltaT*v