117 problems

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…

A macroscale stochastic differential equation models processes arising from microscale dynamics, and the microscale processes may induce a distinction between Itô and Stratonovich…

Let $N$ be odd, and consider the stationary points emerging from the origin at the desynchronisation bifurcation $gamma=gamma_1$. The stationary points have an index, with index-…

Let $N$ be odd, and let $\gamma_1-\delta<\gamma<\gamma_1$ for the positive parameter $\delta=\delta(N)$ from the desynchronisation bifurcation theorem. Let $\cS(\gamma)$ be the set…

Let an SDE system be an additive-noise system satisfying Assumption. d-separation do-calculus conjecture. The $d$-separation analogue of the do-calculus holds for these systems, in…

Let a system of causal stochastic differential equations be a class of SDE systems, and suppose that instantaneous cycles are excluded. d-separation Markov-property conjecture. The…

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…

Consider the stochastic nonlinear differential-equation simulation problem studied in the paper, and let $lambda_i$ denote the dissipation rates ordered increasingly. The strong di…

The models under consideration are stochastic differential equations with dissipative dynamics; stronger dissipation may be induced by an operator such as the standard Stokes opera…

Let $\boldsymbol{\boldsymbol{\rm HP}}_{\mathfrak{s}}$ denote the invariant measure for the hypergeometric-Pearson entire-function process, and let…

Asymptotic consistency conjecture. The property holds as long as

Let $D\sigma\in L_{1,\operatorname{loc}}(\mathbb{R}^{d+1})$ and write $b=b_M+b_B$, where both summands are Borel measurable, $b_M$ is the Morrey part, and $b_B$ is the bounded part…

Let $F$ be a macroscopic observable, and let the coefficients $a$ and $s$ define a stochastic perturbation of $F(\hat{\rho}_{Y_t})$ in the full dynamics referenced in the source. T…

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…

Time-dependent-parameter convergence conjecture. The convergence results should extend to this setting, yielding a bound

Let $d\geq 2$ and let $\beta>0$. Consider a translation-invariant Coulomb interacting Brownian motion in $d$ dimensions, and denote its self-diffusion coefficient by…

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 a stochastic time-periodic Newtonian system be a Newtonian system with periodic coefficients and stochastic forcing, and interpret recurrence through periodicity in distributio…

Commutativity conjecture. The same conditions under which uniform-in-time averaging holds are sufficient to guarantee commutativity of the whole diagram.

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…

Consider a balanced neural network state, meaning a state in which excitatory and inhibitory effects satisfy the balance condition governing the network's mean activity. Algebraic…

Pilipenko's conjecture. The most likely extreme solutions should be those satisfying

Ground-state asymptotic conjecture. As $|x|\to\infty$, one has $\log\psi_1(x)\approx-\rho(x)\approx-U(x)-g(x)$.

Let $b$ and $\sigma$ be the drift and diffusion coefficients of the degenerate stochastic differential equation, and let $a:=\sigma\sigma^*/2$. For an initial measure $\nu$, consid…