Let $G=(G(x))_{x \in \mathbb R^m}$ be a *conservative* random field with values in $\mathbb R^m$, for large positive integer $m$. That is, there exists a scalar random field $g=(g(x))_{x \in \mathbb R^m}$, which depends continuously-differentiably on $x$, such that $G(x) = \nabla G(x)$ for all $x \in \mathbb R^m$. Assume that
- **G is conservative.** 
- **$G$ looks thesame every where.** $G(x)$, $-G(x)$, and $G(x')$ are *equal in distribution* for all $x,x' \in \mathbb R^m$. 
- **$G$ is centered.** $E[G(x)] = 0$.
- **Moment bounds.** There exists $\alpha,\beta,\gamma > 0$ such that for all $x \in \mathbb R^m$,
  - $ E[\|G(x)\|^2] \le \alpha$,
  - $E[\|G(x)\|^4] \le \beta$ (note that this automatically implies the correlation bound $E[\|G(x)\|^2\|G(x')\|^2] \le \beta^2$),
  - $\gamma_1 \le Var[\|G(x)\|^2] \le \gamma_2$. 

Fix $x_0 \in \mathbb R^m$ and consider the differential equation

$$
\begin{split}
\dot{x}(t) &= -G(x(t)), \;t \in \mathbb R\\
x(0) &= x_0.
\end{split}
\tag{1}
$$
For each $t \ge 0$, define $r_t := \|x(t)-x_0\|$ and $c_t := \int_0^t \|G(x(s)\|^2ds$

>**Goal.** My ultimate goal to prove that there is a time $t^\star \ge 0$ (which may depend on dimension parameter $m \to \infty$) such that $\|x(t^\star)-x_0\|$ is small with high-probability and $g(x(t^\star))<0$ with high-probability.

I figured out I could use concentration of measure techniques. This is because, for all $t \ge 0$, $g(x(t)) = g(x_0) - \int_0^t \|G(x(s))\|^2ds$.

>**Question.** How to go about large and small deviation inequalities for $r_t$ and $c_t$ as a function of time $t$ ? More specically, I'd like to how (anti-)concentration inequalities for both $r_t$ and $c_t$.

**Softer question.** I don't know under what topic such problems fall "concentration of random fields" ?, "oncentration of gradient flows" ?, etc. High-level help and references on this point, would already be very useful.

Observations
---
Some parts of my problem can be crudely solved. For example, if $u > 0$, then for all time $0 < t < u/\alpha$ we have

$$
P(c_t > u) \le \frac{E[\int_0^t \|G(x(s))\|^2 ds]}{u} = \frac{\int_0^t E[\|G(x(s))\|^2] ds}{u} \le \frac{t\alpha}{u}
$$

Similarly, a concentration bound for $r_t$ can be obtained using Hoelder's inequality combined with Markov like so

$$
\begin{split}
P(r_t > r) &= P\left(\|\int_0^t G(x(s))ds\|^2 > r^2\right) \le \ldots \le t^2 P\left(\int_0^t \|G(x(s))\|^2ds > r^2\right)\\
& \le \frac{t^2\alpha}{r^2},
\end{split}
$$
valid for all $r > 0$ and time $0 < t < r/\sqrt{\alpha}$.

Completely worked-out example
----
Consider a **static** random field given by $G(x) = w$, where $w \sim N(0,(1/m)I_m)$. Then integrating (1) gives $x(t) = x_0 - tw$. Thus, $r_t = t\|w\|$ and $c_t = t\|w\|^2$, and there is no shortage of anti-/concentration inequalities in this case. We can take $t^\star = \|x_0\|^2/m \cdot e_m$ for any $e_m \to \infty$ and get the following w.p $1-o(1)$

- $r_{t^\star} = \mathcal O(r_m)$.
- $c_{t^\star} = \Theta(r_m)$.
- $g(x(t^\star)) < 0$.