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The so called forgery theorem for Brownian motion says that for any continuous $f: [0, T] \to \mathbb R^d$, with $f(0) = 0$, the $d$ dimensional Brownian motion $W$ has a nonzero chance of staying $\...

Consider the SDI $$dX^\varepsilon(t)\in b(X^\varepsilon(t))\,dt + \varepsilon \sigma(X^\varepsilon(t)) \, dB(t).$$ Is there any Freidlin-Wentzell large deviations principle for $X^\varepsilon$?

Fix $\epsilon>0$ and let $(\Omega,\mathcal{F},\mathcal{F}_t,\mathbb{P})$ be a stochastic base, and let $f:\mathbb{R}^n\to \mathbb{R}^n$ be a continous function with $f(0)=0$. Is there a family of ...