Setting
Suppose I have a discrete dynamical system given by: $$ X^{n+1} = f(X^{n}) \qquad X^0 =x , $$ where $f$ is some diffeomorphism from $\mathbb{R}^{d}$ to itself, and some $x \in \mathbb{R}^d$.
Question
Given a $d$-dimensional Brownian motion $W_t$ and integer $n$.
When does there exist, functions $\tilde{f}:\mathbb{R}^{d\times n}\rightarrow \mathbb{R}^d$, $\rho:[0,\infty)\rightarrow [0,\infty)$ and a predictable-process $u_t$ such that for small $\sigma > 0$: $$ Z_t^{\sigma}= \int_0^t\tilde{f}(Z_{s}^{\sigma},u_s)ds + \sigma W_t \qquad Z_0^{\sigma} = x $$
satisfies the "small deviation-type principle:"
- (Approximation of $X^n$ by $Z_t$) for every $\epsilon>0$ there is some $n_{\epsilon}>0$ satisfying: $$ \limsup_{\sigma \to 0} \mathbb{P}\left\{ \|Z_t^{\sigma} - X^n\|<\epsilon:\, (\forall t \in [n,n+1)) \right\}>\rho(\epsilon) \qquad (\forall n\geq n_{\epsilon}) $$
- (Rate function $\rho$ decays to $0$) $$\lim\limits_{t\to\infty} \rho(t) = 0.$$
Ideally:
Can $\tilde{f}$ be written down explicitly (I don't need $u_t$ to be explicit)?
