Let $x\mapsto g(x)$  be a gaussian random field on $\mathbb R^m$. Let $x_0 \in \mathbb R^n$, and (assuming regularity conditions) consider the gradient-flow

$$
\dot{x}(t) = -\nabla g(x(t)), \;x(0) = x_0
$$

>**Question 1.** Is there a (Kac-)Rice formula for the zero-crossings of $g(x(t)$, namely $\#\{t \in [0, T] \mid g(x(t)) = 0\}$ ?

One of the things I'm interested in are upper-bounds for $g(x(t))$ as a function of $t$.
>**Question 2.** Can one obtain a Borell-TIS bound for the $\sup_{0 \le t \le T} g(x(t))$.

**Note.** I'm not necessarily looking for a clear-cut answer (though this would be really cool), but general guidelines on how to go about this.

Examples
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As a working examples, one could consider the fields
- **Linear gaussian random field** wherein $g(x) = w^Th(x)$, with $w \sim N(0,I_k)$ and deterministic $h \in \mathcal C^1(\mathbb R^n \to \mathbb R^k)$.
- **Stationary gaussian random fields**