Random linear differential equations

Definitions and mathematical statement

For each integer $n\ge1$, let $A_0,\ldots,A_n$ be independent real normal random variables of mean zero and variance one. With probability one $A_n\ne0$, so \[ P_n(z)=A_nz^n+A_{n-1}z^{n-1}+\cdots+A_0 \] has degree $n$. Define \[ p_n=\mathbb P\{\operatorname{Re}z<0 \text{ for every root }z\text{ of }P_n\}. \] Roots are tested with their usual complex multiplicities, although only their locations matter for the event. Strict negativity excludes roots on the imaginary axis.

Equivalently, $p_n$ is the probability that the zero solution of the constant-coefficient equation $A_nx^{(n)}+\cdots+A_1x'+A_0x=0$ is globally asymptotically stable when the state consists of $x,x',\ldots,x^{(n-1)}$. The question has two parts: determine an asymptotic expansion of $p_n$ as $n\to\infty$, and decide whether $p_{n+1}<p_n$ for every $n\ge1$.

The distribution is exactly the independent identically distributed Gaussian coefficient model above. Conditioning a leading coefficient to equal one, choosing random roots directly, or taking independent entries of a random matrix gives different probabilities. An asymptotic result should specify its scale and error term, distinguishing a logarithmic decay estimate from a full leading equivalent or higher-order expansion. Strict monotonicity is a separate assertion for all degrees and would not follow merely from eventual asymptotic decay.

Short English statement

Find the asymptotic probability that a Gaussian-coefficient polynomial is Hurwitz stable, and decide whether it strictly decreases with degree.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4700021 (AMR-046-0021), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Gasull - Some open problems in low dimensional dynamical systems (2020), Problem environment 21: Random linear differential equations. Original source indexed by AMR-046-0021. https://arxiv.org/abs/2012.02524.