I'm doing some work on stochastic processes, where I use random functions which are defined by their properties over averages, i.e $$\langle f(t) \rangle =0 \\ \langle f(t) f(t')\rangle = \alpha(x,y) e^{-(t-t')^2}$$ Is there a way to define something like this?
My problem is that I have a multitude of spatial derivatives applied to those functions in different ways which I can take out of the time average i.e. $$\langle \partial_x f(t) \partial_{y'} f(t')\rangle = \partial_x \partial_{y'} \langle f(t) f(t')\rangle$$ Is there a way to just do the algebra with the operators where I could then apply a complex differential operator onto my correlation function?
Best