Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The mean Euler characteristic and excursion probability of Gaussian
  random fields with stationary increments   Let $X=\{X(t),t\in {\mathbb{R}}^N\}$ be a centered Gaussian random field with
stationary increments and $X(0)=0$. For any compact rectangle $T\subset
{\mathbb{R}}^N$ and $u\in {\mathbb{R}}$, denote by $A_u=\{t\in T:X(t)\geq u\}$
the excursion set. Under $X(\cdot)\in C^2({\mathbb{R}}^N)$ and certain
regularity conditions, the mean Euler characteristic of $A_u$, denoted by
${\mathbb{E}}\{\varphi(A_u)\}$, is derived. By applying the Rice method, it is
shown that, as $u\to\infty$, the excursion probability
${\mathbb{P}}\{\sup_{t\in T}X(t)\geq u\}$ can be approximated by
${\mathbb{E}}\{\varphi(A_u)\}$ such that the error is exponentially smaller
than ${\mathbb{E}}\{\varphi(A_u)\}$. This verifies the expected Euler
characteristic heuristic for a large class of Gaussian random fields with
stationary increments.

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The categories of this paper are: q-bio
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-lat
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The categories of this paper are: quant-ph
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
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The categories of this paper are: eess
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: cond-mat