820 problems

Let $\lambda(\alpha,\beta)$ denote the eigenvalue process defined by the paper's Dyson SDE, with interaction parameter $\alpha$ and confinement parameter $\beta$. For the time-resc…

Let $(U_t)_{t\geq 0}$ and $(G_t)_{t\geq 0}$ denote the unitary and general linear multiplicative Brownian motions, respectively. The process $(U_t)_{t\geq 0}$ converges in $*$-dist…

Pointwise convergence conjecture. This replacement should make the field converge pointwise. The proposed summation is expected to average out the fluctuations caused by the locall…

Mean-field equation conjecture. The dynamics of $\mu_t^*$ is given by $\mathcal L(X_0^*)=\lim_{N\rightarrow\infty}\mathcal L(X_0^{1,N})=\mu_0^*$, with $S_0^*$ distributed according…

Product-form characterization conjecture. The network $(\mathcal{G},\kappa)$ has a product-form stationary distribution if and only if it is complex balanced.

Continuity conjecture. The map

Let $X_t$ be the one-dimensional Brownian polymer process with interaction function $f$. Assume that the interaction is symmetrical and repulsive, with limited interaction, namely…

Let the frog model start with one frog per vertex on the homogeneous tree $\mathbb{T}_d$ of degree $(d+1)$, with frogs performing the model's random walks and waking sleeping frogs…

Let $\mathcal W$ be a renormalization class, let $F^{(n)}w$ be its iterated renormalizations, and let $K^{w,(n)}$ be the corresponding iterated probability kernels. Assume that…

Let $D\subset\mathbb R^d$ be the domain of a renormalization class $\mathcal W$, let $(c_k)_{k\geq0}$ be positive migration constants, and define $s_n=\sum_{k=0}^{n-1}c_k^{-1}$ and…

Let $D_\lambda$ denote the diffusion coefficient measured at scale length $\lambda$ in turbulent air. Richardson's conjecture. The diffusion coefficient $D_\lambda$ depends on the…

Let $\bm W=(\bm W(t):t\geq 0)$ be planar Brownian motion and let … be the convex hull of its path up to time $t$. Lévy's smoothness conjecture. The boundary of $\mathcal{H}(t)$ sho…

Consider a two-type SIS model in which the two species have general, possibly unequal, death rates and compete for space. For each species, define its birth/death ratio as its birt…

Residual diffusivity conjecture. If the deterministic flow of $u$ on the torus is chaotic, then $D^\epsilon_\mathrm{eff}$ is $O(1)$ as $\epsilon\to0$.

Let $Y$ be the multivariate fractional Ornstein–Uhlenbeck process with component Hurst parameters $H_1,\ldots,H_d$, and suppose that the moving-average representation is known when…

Let $\mathbb{S}$ be a compact subset of $\mathbb{R}^d$, and let ${\sf L}^\rho$ denote the operator associated with the filter stability framework. Compactness conjecture. The opera…

Let $a_t$ be an independent and identically distributed process, and let $r_i$ follow either a linear regression model with white noise or a weighted random-walk model. Let $\pi_5$…

KPZ universality conjecture. All KPZ models, under the so-called $1:2:3$ scaling, converge to the KPZ fixed point.

Let $\rho_t$ denote a super-Brownian motion and let $L_x$ be its total occupation-time density, defined by … for suitable test functions $\phi$, in dimensions $d\leq 3$. The Hölder…

Let $Z_t$ be reflected Brownian motion in the generalized parabolic domain ${\mathcal D}$, let $X_t$ denote its horizontal coordinate, and define … where $b$ describes the domain's…

Non-Gaussianity conjecture. The term $I_i(t)$ is not Gaussian.

Stationary-measure conjecture. The process $(\widetilde H_t)_{t\in[0,\tau]}$ represents the stationary measure of the KPZ equation on the interval $[0,\tau]$ with Neumann boundary…

Stationary-distribution conjecture. For each $\theta>1$, the process $Z_t$ admits exactly two stationary distributions: the trivial one, the Dirac mass at $0$, which is reached onl…

Let $L_{t}^{s,x}$ be the local time of $\varphi(s)$ at level $x$. Let $(\mathcal{P}_{y})_{y\geq0}$ be the semigroup and let $\mathcal{P}^{+}_{x}$ and $\mathcal{P}^{-}_{-x}$ be the…

Poisson Hypothesis. Spiking deliveries to distinct replicas shall be asymptotically distributed as independent Poisson (or compound) point processes.