71 problems

Let $f$ satisfy $f(0)=0$ and $f'(0)>0$. For the characteristic equation at $x=0$, … write $\theta=rf'(0)-\gamma$ and let $W_m$, $m\in\mathbb Z$, denote the $m$-th branch of the Lam…

Turbulence is considered in the setting of the Boussinesq equations and their Lagrangian process, where deterministic and stochastic descriptions of the fluid dynamics are compared…

Let $\psi_r$ and $\psi_i$ be the quadratic noise processes arising in the refined stochastic amplitude equations for the Duffing–van der Pol oscillator, and let $\phi$ denote the o…

Let such measures be invariant measures of systems with a stochastic dynamics, and suppose that they have no detailed balance. In the spin-flip setting, consider non-reversible pro…

Let $\nu$ be the invariant measure in Corollary for the stochastic Wright equation, and let $\sigma_0$ denote the noise parameter. Uniqueness conjecture. In the presence of noise,…

Stochastic embedding sufficiency conjecture. The bound

Let $(\rho_t)_{t\geq 0}$ be the curve of marginal laws associated with the controlled swarm gradient dynamics, and let $\rho_\infty$ denote its limiting measure. Let $\beta(t)$ be…

The Lorenz system is perturbed by strong noise, and its top Lyapunov exponent is the largest Lyapunov exponent associated with the resulting stochastic dynamics. Lorenz synchroniza…

Generic constancy conjecture. Suppose that $\Pc_{\mathrm{in}}^-$ and $\Pc_{\mathrm{state}}^-$ are Polish and that…

The stochastic model has online transmission rate $\tau$, offline recovery parameter $\gamma_p$, offline parameter $\eta$, and offline-to-online feedback strength $\theta$. Conside…

Let $x^{(\epsilon)}_{(\delta)}$ be the forced subsystem driven by a chaotic forcing, with a chaotic subsystem satisfying moderate assumptions and with sufficiently large time-scale…

The paper considers reaction-diffusion equations of the form … where $A$ and $f$ are the operator and reaction term from the deterministic equation, and $N$ is a Hölder continuous…

Consider the stochastic nearest-neighbor Kuramoto model with interaction range $r>1$, system size $n$, twisted states indexed by $q$, prefactor $C(q,n)$, and energy barrier…

Let $\balance_\mathbf{G}$ denote a $\mathbf{G}$-balanced manifold, with $\balance=\balance_{\mathbf{0}}$ the balanced variety, and consider random-Langevin evolution (RLE) in the r…

Let $\nu_0$ and $\nu_1$ be endpoint measures, and let $\operatorname{UOT}^{\mathrm{D}}(\nu_0,\nu_1)$ be the diffusion-based unbalanced optimal transport functional defined by minim…

Let a stochastic oscillator have real eigenfunctions associated with more rapidly decaying modes, and let the corresponding eigenvalues be compared with the leading negative eigenv…

Let $V$ be decomposed into a resolved effective potential $U_k=V_0+\sum_{j=1}^{k}V_{j,\varepsilon_j}$ and under-resolved microscopic components…

Consider the stochastic version of the Brusselator with the noisy perturbation formulated in the cited stochastic model. In the deterministic Brusselator with large bifurcation par…

Let the Brusselator be subjected to a noisy term representing a stochastic variation of its bifurcation parameter. Parametric-noise destruction conjecture. Parametric noise destroy…

Let inertial gravity wave amplitudes evolve in geophysical flows forced by turbulent eddy fields, and consider their mean square displacement over long times. Linear growth conject…

Heterogeneous-network VFPE conjecture. For heterogeneous networks, the Vlasov–Fokker–Planck equation associated with the stochastic network should be

Let $\bm L_{P}$ be the unweighted Laplacian matrix of the Petersen graph, which is regular with degree sequence $\bm d=\operatorname{diag}(\bm L_{P})=(3,3,\ldots,3)$. Let $\bm L^*$…

Let $w$, $v$, $d$, and $\beta$ satisfy … and let $0<\bar{\epsilon}<1$. Suppose the ODE solution trajectory is increasing below $\bar{\epsilon}$ and decreasing above…

Trajectory-class decomposition conjecture. Natural decompositions of complex thermodynamic processes into their mesoscopically distinct trajectory classes generally provide traject…

Let $\rho$ be a positive integer, let $\bar{\lambda}$ be the average parameter value, and let $p_i(\bar{\lambda})$ for $1\leq i\leq 2^\rho$ denote the stable period-$2^\rho$ points…