10 problems

Let $N$ be a positive integer and let ${\theta}(k:l)$, for $k,l\geq 0$, be the parameters defining the generator ${\mathcal A}_N^\theta$; the boundary values ${\theta}(k:0)$ and…

Skoulakis–Adler conjecture. The conditional log-Laplace transform of $X_t$ should be the unique solution to a nonlinear stochastic partial differential equation.

Consider the stochastic flow on $\mathbb{R}^2$ generated by the Hopf normal form with additive noise … Let $\lambda$ be the Lyapunov exponent of this system, with $b$ the shear par…

KPZ tail-fluctuation conjecture. The fluctuations of the random field $P^\omega(r,y)$ should be described by the KPZ equation at some point within the tail of that probability meas…

The KIW formula describes the stochastic evolution of a time-dependent $k$-form under stochastic transport, with its final expression being coordinate free. A manifold is a space o…

Consider the time-dependent cellular flow defined by Eq., with parameter $B=2.72$, and let $D^E$ denote its effective diffusivity matrix. Vanishing-viscosity diffusivity conjecture…

Linear-growth conjecture. A drift with linear growth, together with globally Lipschitz noise, implies exponential growth of the image of a bounded set, or at least non-explosion.

The uniform-bound conjecture. The following bounds should hold for every $m\geq 0$ and for $u$:

Let $\pi^0\in{\cal W}_0$ be a path and let $\rho_t(\pi^0(t))$ denote the mass associated with its position at time $t$. Mass-variation conjecture. The function … has locally unboun…

Let $X$ be a compact set and let a stochastic flow act on $X$. Its image at time $t$ is denoted by $\varphi_t(X)$, and $\operatorname{diam}$ denotes diameter. Carmona's conjecture.…