# Stochastic Processes – Is the probability space really an important part of a weak solution to an SDE?

A weak solution is a tuple $$( Omega, mathcal F, mathbb P, X, B, ( mathcal F_t))$$ with $$( Omega, mathcal F, mathbb P)$$ a probability space, $$B$$ a $$( mathcal F_t)$$Brownian motion and $$X$$ a $$( mathcal F_t)$$– adapted process satisfying certain regularity conditions and of course the given SDE.

In the two examples that I saw, we always chose a probability space $$( Omega, mathcal F, mathbb P)$$ which is rich enough to admit Brownian motion.

So, can not we just use the same space each time?