Please ignore the scary length of the proof! I only query two lines!
So, I have two questions about the following part of the theorem. It is a paper looking at Stochastic Partial Differential equations, and here is an effort to bound the noise term, which essentially an Ornstein-Uhlenbeck process. I am very new to working with noise, so I am unsure of the following
- What does $\beta (\gamma(T))$ mean? $\beta$ is described as a Brownian motion (I know what this is at least!), but I have no idea what this means in terms of an expression when it is applied to a function.
- Also, when mentioning Doob's Theorem, I was not too sure what occurred. I took it to mean Doob's decompostion theorem, but although I understand this intuitively, I cannot see how it is applied here.
Help would be appreciated!!!