37 problems

Let $(\mathcal{M},g)$ be a Riemannian manifold, let $A$ be a smooth vector field, and let $\mu$ be the loop measure determined by the diffusion with generator … Assume that the dif…

Fix $\theta\in\mathbb{R}$, and let $utheta$ be the invariant measure associated with the solution $V^{(\theta)}$ of the SPDE started from the constant initial profile $\theta$. Assu…

Consider an annealed time-ergodic invariant measure for the SPDE discussed in the paper. A random field is weakly spatially stationary when it has the weak spatial stationarity pro…

Let $utheta$ denote the annealed invariant measure obtained from the constant initial profile $thetariinmathbb{R}$, and let and denote, respectively, the stronger and baseline wea…

Let $K\subset\mathbb{R}$ be a nonrandom compact set, let $\mathfrak S(\theta)$ be the random set of points of slow growth at parameter $\theta$, and let $\lambda(\theta)$ denote th…

Let $\theta_c$ be the critical parameter and let $\mathfrak S(thetac)$ denote the corresponding random set of points of slow growth. Write $\operatorname{\dim{{rm H}}}$ for Hausdorf…

Multiplicative-noise extension conjecture. Theorem (ii) and (iii) for the additive-noise equation remain valid in the presence of multiplicative noise, provided that

Uniform dimension conjecture. The uniform dimension theorem stated for $p\geqslant4$ is valid whenever $p\geqslant2$, provided that

Metastable random-walk conjecture. As $\epsilon\to0$, the rescaled process approaches a symmetric simple random walk on

Consider the 3D incompressible Euler equations on $[0,T]\times \mathbb{T}^3$. A weak solution is understood in the usual distributional sense, and $C^{\beta}$ denotes the Hölder sp…

Effective diffusivity conjecture. The sequence $u^N$ converges weakly to the stationary solution $u$ of

Let $U\subset\mathbb{R}^d$ be a bounded domain satisfying the hypotheses of either Theorem or Theorem, and let $u(t,x)$ be the corresponding solution to the stochastic heat equatio…

Let $\mathcal{H}_{u,v}$ denote the explicitly constructed stationary processes for the half-space KPZ equation with Neumann boundary parameter $u$, indexed by…

Projected Wiener dynamics conjecture. For $T_\varepsilon\leq c\varepsilon\eta_\varepsilon^{-1}\wedge\tau$, one expects

For $\eta,\epsilon>0$, let $(u_t^{\eta,\epsilon})_{t\geq 0}$ be the stationary $K$-valued Markov process associated with the Dirichlet form $\mathcal{E}^{1,\eta,\epsilon}$ and star…

Let $K=\{h\in H:h\geq 0\ \text{a.e.}\}$, where $H=L^2(0,1)$, and let $\mu$ be the law on $K$ of a reflecting Brownian motion on $[0,1]$. Let $(u_t)_{t\geq 0}$ be the reversible Mar…

Let $d\geq 3$, let $h_{x_i}=h_0\cdot 2^{-l_i}$ for $i=1,\ldots,d$, and let $\Delta=\Delta_1\otimes\cdots\otimes\Delta_d$ be the tensor product of the first-order difference operato…

Let $d\geq 3$, and let $v$ solve the $d$-dimensional Zakai SPDE. For a spatial grid with spacings $h_{x_i}>0$, let $V_{i_1,\ldots,i_d}^N$ denote the implicit finite-difference appr…

Curvature coefficient conjecture. The universal constant is

Consider the stochastic Allen–Cahn equation driven by a multiplicative infinite-dimensional ${\bf Q}$-Wiener process, with the drift-implicit Euler–Galerkin scheme and numerical ap…

Let $u$, $\lambda$, and $D$ satisfy the effective-coupling condition from the paper, and let $(h^\varepsilon){\varepsilon0}$ denote the rescaled KPZ fields, with…

Let $(\sigma_k)_{k\in\mathbb{Z}^2_0}$ be the family of vector fields on $\mathbb{T}^2$ defined, for $\gamma\geq 2$, by … Assume that the associated gradient estimate is the estimat…

Let $d\in\mathbb{N}$ and let $F_\omega(t,x)$ have exponential decay of space-time correlations. For every $b\in\mathbb{R}^d$, consider global solutions of the random Hamilton–Jacob…

Spatial asymptotic conjecture. In the particular case of space-time white noise,

Global infinite-oscillation conjecture. There exist such correlation functions $f$ for which