19 problems

Hörmander nondegeneracy conjecture. If the parabolic Hörmander condition holds everywhere, then it cannot be the case that $b(x)\in\operatorname{Im}\sigma(x)$ for any $x$; equivale…

Consider the random dynamical system induced by the stochastic differential equation, and fix $a>0$ and $\beta\in\mathbb{R}$. Let $\Lambda_1$ denote its largest Lyapunov exponent.…

Let $u_0$ be an observable depending only on the first $k$ coordinates, so that … Let $\lambda_i$ be dissipation rates satisfying $\lambda_i\propto i^p$ for an integer…

Let $M_+$ be the interior state space of the stochastic Rosenzweig–MacArthur model, let $x_1^x(t)$ denote the first component of the solution started from $x\in M_+$, and let…

Let $\mathbb{X}^{\circ}$ denote the interior of the state cube, let $\mathbb{F}$ be the collection of faces, and let $F\in\mathbb{F}$ be an attracting face supporting the invariant…

Stochastic differential approximation conjecture. The process $\mathcal{X}_n$ behaves like a continuous-time process $X_t$ satisfying

Let the assumptions of Theorem hold. For $\epsilon>0$, let $(\bm{X}^{(\epsilon)}(t),\bm{w}^{(\epsilon)}(t))$ solve, and let $(\bm{X}^{(0)}(t))$ solve the limiting system with the s…

Let the assumptions of Theorem hold, with the dynamics of $(\bm{X}_i(t))_{i\in[N]}$ and $(\bm{w}_{i,j}(t))_{i,j\in[N]}$ governed by the SDEs with weight-noise intensity $\eta>0$. L…

Let $u_0\in L^2_\sigma$ be divergence-free initial data, let $u$ be an associated Leray solution of the three-dimensional Navier–Stokes equations, and consider the stochastic diffe…

Let $b$ be a finite signed measure on $\mathbb{R}$, let $H\in(0,1/2)$, and consider the stochastic differential equation … where $B^H$ is fractional Brownian motion with Hurst para…

Convergence-rate conjecture. There is a constant $H$ independent of $\Delta$ such that

Continuity and propagation conjecture. The process $X$ has a continuous transition density function $p(t,x,y)$. Moreover, if

Let $x_0\in\mathbb{R}$, let $\mu,\sigma\colon\mathbb{R}\to\mathbb{R}$ satisfy (A1)–(A3), and let $X$ be the solution of the corresponding SDE. For $n\in\mathbb{N}$, observations co…

Let $X$ solve the quasilinear Skorohod SDE … where $X_0\in L^p(\Omega)$ for some $p>2$, $a$ satisfies the global Lipschitz and boundedness condition … for all $t\in[0,1]$,…

Let $f:{\mathbb{R}}^d\longrightarrow {\mathbb{R}}^m$ be continuous and piecewise Lipschitz with exceptional set $\Theta$. Suppose that $f$ is intrinsically Lipschitz with respect t…

Extinction conjecture. If $R_0^S<1$ and

Let $\beta$ and $Q$ be arbitrary $d\times d$ matrices, let $p<\frac{d+1}{2}$, and let $X_t$ be a solution of the Wishart stochastic differential equation with initial condition…

Let $T\in(0,\infty)$, let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space with a normal filtration $(\mathcal{F}_t)_{t\in[0,T]}$, let $W^{l,k}$ be a family of independent…

Let $\left(\overline{\mathcal R_{[L]}}\right)_{L=1,2,\dots}$ be a sequence of restriction operators whose macroscopic state variables are the first $L$ centralized moments, with…