  We consider the first exit point distribution from a bounded domain $\Omega$
of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevin
dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from
deterministic initial conditions in $\Omega$, under rather general assumptions
on $f$ (for instance, $f$ may have several critical points in $\Omega$). This
work is a continuation of the previous paper \cite{DLLN-saddle1} where the exit
point distribution from $\Omega$ is studied when $X_0$ is initially distributed
according to the quasi-stationary distribution of $(X_t)_{t\ge 0}$ in $\Omega$.
The proofs are based on analytical results on the dependency of the exit point
distribution on the initial condition, large deviation techniques and results
on the genericity of Morse functions.
