What is Stochastic differential equation, strong and weak solutions?

A stochastic differential equation is

for measurable $\mu, \sigma$. A strong solution on a given probability space with a given Brownian motion $W$ is an adapted continuous process with $X_t = x_0 + \int_0^t\mu(s, X_s)\,ds + \int_0^t\sigma(s, X_s)\,dW_s$. A weak solution is a pair $(X, W)$ on some filtered probability space satisfying the equation: the Brownian motion is part of the answer.