4 problems

For $\eta>0$, consider the coupled transport–diffusion system with state variables $y$ and $\hat y$ on $[0,1]$: … with boundary conditions … Let…

Weak-limit conjecture. There exists a probability measure $\mu_{\infty}(\alpha)$ on $\mathbb R$ such that, as $t\to\infty$,

Boundary scaling limit conjecture. For every $\alpha\in(0,1)$, there exists $\ell_{\alpha}>0$ such that

Let $n\geq 1$ and let $\gamma_n$ be the standard Gaussian probability measure on $\mathbb R^n$. For $t>0$, let $U_t$ denote the Gaussian diffusion operator, and for a measurable no…