Being trained as a mathematician I am trying to understand stochastic field theory (i.e. field theory applied for dynamics of stochastic processes, e.g. SDE) and I have a difficulty at one point.
Suppose I have an Ornstein-Uhlenbeck process $$ \dot{x}(t) + ax(t) - \sqrt{D} \eta(t) = 0$$ which has probability density $$ P(x) = \int \mathcal{D} \tilde{x} e^{S[x,\tilde{x}]} ,$$ $$S[x,\tilde{x}]=\int \left( \tilde{x}(t)\bigl(\dot{x}(t)+ax(t)-y\delta(t-t_0)\bigr) -\frac{D}{2} \tilde{x}^2(t) \right) dt $$
The variable $\tilde{x}(t)$ is called auxiliary or 'response' variable (for reasons that I don't understand, unfortunately). In several sources I saw written that linear response has the form $\langle x(t),\tilde{x}(t')\rangle .$
Question: why does the (quite artificial) auxilary variable has something to do with linear response? In my understanding the words 'linear response' mean the size of the system $x(t)$ response to a perturbation, so I don't understand both 1) for what reason $\tilde{x}$ appears there 2) why the formula has a form of the second moment, instead of someting like $\langle \frac{\delta x(t)}{\delta\eta(t)} \rangle$.
More precisely I am trying to understand this article.
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