What is a general Brownian Motion?

Question

This might be a dumb question, but no textbook ever defines what a "Brownian Motion" is, just what a "Standard Brownian Motion." I always assumed that a Brownian Motion is any random variable that can be represented at $\mu t+\sigma B_t$ where $B_t$ is a Standard Brownian Motion, but now I am not sure due to how this would affect the covariance function. What is the generally accepted definition?

Answer 1

You're right, it seems to depend largely on sources.

In these notes, the (general) Brownian motion is defined as $\sigma B_t$ where $B_t$ is a standard Brownian motion. And what you describe is called a Brownian motion with drift.

See also here.

Sometimes the Brownian motion with drift is said to be $x+\mu t+\sigma B_t$.

But the most important is to know the definition of a standard brownian motion, its characterizations and properties. Then you can determine how these properties are affected when you transform it. Let's take your example of a drifted brownian motion $X_t=\mu t+\sigma B_t$. It's still a gaussian process, with stationary and independent increments, almost surely continuous, with $X_t\sim\mathcal{N}(\mu t,\sigma^2t)$ and $Cov(X_t,X_s)=\sigma^2 Cov(B_t,B_s)=\sigma^2\min(t,s)$.