I have not done very much stochastic calculus but know the basics. (Ito formula, Ito lemma, Stochastic Integral construction, the properties of B.M, Some properties of the stochastic integral, I have seen the usual existence and uniqueness of solutions to SDE with Lipshitz condition as presented in Oksendal's stochastic calculus book.)
I have been given some reading about Krylov-Bogoliubov Method for constructing invariant measures.
An SDE in Hilbert space $H$ is introduced as
$$d(X)=b(X)dt + \sigma(X)dW $$
With $W$ a cylindrical B.M in the Hilbert space $\mathbb{H}$. The SDE is Lipshitz in the following sense
$$|b(X)_{1}-b(X_{2})|_{H}-|\sigma(X_{1})-\sigma(X_{2})|_{L_{2}(\mathbb{H},H)}\leq C|X_{1}-X_{2}|_{H} $$
Then the author claims that a unique pathwise solution exists, what do they mean by a pathwise solution? By a solution all I can think of is a stochastic process $X_{t}$ which satisfies the SDE.