I was just reading an article in which the authors simulate an equation with added noise based on an Ornstein-Uhlenbeck process. This is the equation:
They describe the term $ n(t) $ in equation A1:
I would like to reproduce this equation, but don't entirely understand what the authors did there.
Firstly, this does not look to me like the definition of an Ornstein-Uhlenbeck process (at least as far as I know)
Secondly, can somebody explain to me, what this last term (angle brackets, representing the white noise process) means? I know angle brackets from the inner product or Expectation values and it seems to me like a DiracDelta impulse at time $ t'$, but I don't understand what this $ t'$ means.
To whom this makes more sense, any help appreciated ;)!
For the first part, the stochastic equation of the Ornstein-Uhlenbeck process is often rewriten as a Langevin equation in physical sciences, which is done here (see e.g. [1]). For the second part, $\left< \xi (t)\xi (t')\right>$ denotes the autocorrelation function of a stationnary random process (here, a white noise). By definition of white noise, the later is a Dirac delta, since its Fourier transform -- the power spectral density -- is constant (see e.g. [2]). The notation with brackets for expected values can be found in quantum physics.