# SDE of a standard Brownian motion - Langevin equation

Assume we have a standard Brownian motion $$W_t$$ as the solution to the following SDE

$$dX_t=\mu dt+\varepsilon dW_t$$

1. Which kind of SDE it is? Ito process? For which kind of processes we can say that the drift coefficient and $$\varepsilon$$ does not depend on time or $$X$$? I assume that in the standard Brownian motion the increments are independent and do not depend on time. So, why in the literature mostly the coefficients depend on time and $$X$$?

2. How can i relate this SDE to the Langevin equation governed on the movement of a massless particle immersed in a flow(Brownian motion)

$$0=-\gamma \dot x+f(t)$$

• Such $X_t$ is sometimes also called "Arithmetic Brownian Motion".
– fes
Aug 15, 2020 at 11:34

Your question is rather unclear. First, the Brownian motion doesn't solve $$\,\mathrm d X_t=\mu\,\mathrm d t+\varepsilon \,\mathrm d W_t,\quad \mu\in\mathbb R, \varepsilon >0.$$ A solution of such equation is called Brownian motion with drift. Moreover, you can find a closed form of the solution : typically $$X_t=X_0+\mu t+\varepsilon W_t,\quad t\geq 0.$$