# Probabilistic Approximation of non-linear Dynamical System by Diffusion Process

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)?

• I made some edits since there was no explicit need for the control $U^n$ in the discrete system. – AIM_BLB Jan 23 at 8:00
• You're right, I didn't need the control. Thanks! – AnnieTheKatsu Jan 23 at 8:01