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

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

Integrating the above system gives $$ g(x(t)) = g(x_0) + \int_0^t \langle \nabla g(x(t)),\dot{x}(t)\rangle dt = g(x_0)-\int_0^t\|\nabla g(x(t))\|^2 dt. $$

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

As a working examples, one could consider the following "simple" 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