I am really not a probabilist and I apologize if my question is too naive or not appropriate, please feel free to migrate to SE.
A bit of context: usually, Schilder's theorem tells us that the brownian motion with diffusivity $\nu>0$ (or rather, the Wiener measure $R^\nu$ on $\Omega=\{\omega\in C([0,1];\mathbb R^d)\}$) started from the origin satisfies a large deviation principle as $\nu\to 0$ with scale $\nu$ and rate function $A(\omega)=\frac 12\int_0^1 |\dot\omega_t|^2 dt$ (if $\omega$ is $H^1$ and $\omega_0=0$, and $A(\omega)=+\infty$ otherwise).
This however stands for the Brownian motion pinned at the origin (or at any other point, for that matter). What happens if one whishes to also pin the motion at the other endpoint $t=1$, in addition to $t=0$?
My specific question:
For fixed $x,y$ is there a Schilder theorem for the (law of the) Brownian bridge $R^{\nu,x,x}=R^\nu(\,\cdot\,|X_0=x,\,\cdot\,|X_1=y)$?
I suspect that the anser is yes, except that the rate function should be now $ A^{x,y}(\omega)=\frac 12 \int_0^1|\dot\omega_t|^2dt -\frac 12 |\omega_1-\omega_0|^2 $ if $\Big[\omega\in H^1$ with $\omega_0=x,\omega_1=y\Big]$, and $A^{x,y}(\omega)=+\infty$ otherwise. I am aware that this is ambiguous, since the conditionned $R^{\nu,x,y}$ is only defined for $R^{\nu}_{0,1}$-almost all $x,y$. (and in fact my real problem is set on the flat torus $\mathbb T^d$, and the Wiener measure I'm talking about is really the reversible one, i-e started from the stationary Lebesgue measure)
Note: I can only suspect that this is a possible way to recover the fact that the bridges $R^{\nu,x,y}$ converge to deterministic geodesics $[x\to y]$, since indeed $A^{x,y}$ is minimized by the unique geodesic $\omega^{x,y}_t=(1-t)x+ty$?
Strangely enough I could not find anything in this direction anywhere in the literature, including in the classical textbook of [Dembo & Zeitouni] on large deviations. Any help will be greatly appreciated.
