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I still am novice as far as probability is concerned and after fruitlessly Googling for an answer for a few days I thought I might have a better chance with MO.

Let me first formulate the question/claim. $\newcommand{\bR}{\mathbb{R}}$ $\DeclareMathOperator{\ev}{\mathbf{ev}}$ $\newcommand{\bu}{\boldsymbol{u}}$ $\newcommand{\bsU}{\boldsymbol{U}}$

Fix positive integers $N\geq n$ and suppose that $\bu$ is a Gaussian random map $\bR^N\to\bR^n$ satisfying the following conditions.

Regularity. The random map is a.s. $C^\infty$.

$0$-jet ampleness. For any $x\in \bR^N$ the $\bR^n$-valued Gaussian random variable $\bu(x)$ has a nondegenerate covariance form. Equivalently this means that its probability distribution $\Gamma_x$ is absolutely continuous with respect to the Lebesgue measure on $\bR^n$. Another, more geometric, way of phrasing this is that $\Gamma_x(V)=0$ for any proper vector subspace of $\bR^n$.

I claim that under the above assumptions one can conclude that $0\in\bR^n$ is almost surely a regular value of $\bu$.

Here is some evidence backing this claim.

I. The conclusion is true in the case $N=n$. One source for this result is Proposition 6.5 in the book

Level sets and extrema of random processes and fields by Azais and Wschebor, Wiley, 2009.

The earliest reference on this type of question seems to be a paper by Bulinskaya in the 60s. Let me mention that in Proposition 6.12 in the above book the a.s. transversality is proven for any $N\geq n$ when the $0$-jet ampleness condition is replaced by a stronger requirement I would call $1$-jet ampleness. This condition imposes additional restrictions on the differential of $\bu$. I would rather not involve the derivatives of $\bu$.

II. The conclusion is true for a large class of random maps constructed as follows.

Fix a finite dimensional subspace $\bsU\subset C^\infty(\bR^N,\bR^n)$. We assume that $\bsU$ is ample, i.e., for any $x\in \bR^N$ the evaluation map

$$\ev_x:\bsU\to\bR^n,\;\;\bsU\ni \bu \mapsto \bu(x)\in\bR^n $$

is onto. Next fix an inner product $(-,-)$ on $\bsU$ and denote by $\Gamma$ the Gaussian measure on $\bsU$ determined by this inner product. An element $\bu\in\bsU$ is thus a smooth random map $\bR^N\to\bR^n$. The ampleness of $\bsU$ implies the $0$-jet amplenes condition on $\bu$.

One can verify rather easily that $0$ is .s. a regular value of this random map $\bu$. The argument goes as follows. Consider the incidence set $\newcommand{\eR}{\mathscr{R}}$

$$\eR=\bigl\lbrace\; (x,\bu)\in \bR^N\times \bsU;\;\;\bu(x)=0\;\bigr\rbrace. $$

The ampleness of $\bsU$ implies that $\eR$ is a smooth manifold of dimension $(N-n)+\dim\bsU\geq \dim\bsU$. We next observe that if $\bu$ is a regular value of the natural map

$$\pi:\eR\to\bsU,\;\;\eR\ni(x,\bu)\mapsto \bu\in \bsU $$

then $0$ is a regular value of $\bu$. Sard's theorem applied to the smooth map $\pi$ implies that $0$ is a.s. a regular value of $\bu$.

It is very likely this question has been investigated by people working on random maps. As I said in the beginning I was not able to find references on this question. Any additional info you might have is greatly appreciated. Thank you in advance!

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