171 problems

Consider the nonparametric estimation problems for the space-dependent diffusivity, transport, and source coefficients in stochastic evolution equations described above, with the r…

Let $u=u(t,x)$ be defined for $(t,x)in\mathbb{R}_+\times\mathbb{R}$, let $\beta\in(0,1)$ and $\alpha\in(0,2)$, and let $\dot{W}$ denote space-time white noise. Consider the space-t…

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

Let $u(t,x)$ solve the one-dimensional parabolic Anderson model with space-time white noise $xi$ and narrow wedge initial condition: … For fixed $t>0$, Huang–Lê's spatial asymptoti…

White-noise evolution-system conjecture. The family $\{\mu_{1+t}\}_{t\in\mathbb{R}_+}$ of white-noise measures with variance $1+t$ is an evolution system of measures for the stocha…

Moment upper-bound conjecture. The moment upper bound can also be established for spatially colored noise.

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

The $t^{1/3}$ superdiffusivity conjecture. In particular, there is a constant $C$ such that

Covariance-independent regularity conjecture. With suitable improvements of some of the arguments given in the paper, one can prove continuous dependence in a suitable space of reg…

Consider the stochastic Burgers equation … on $L^2([0,1])$, with viscosity $\nu>0$, under the hypotheses on the coefficients stated for equation (4.11). Burgers stability conjectur…

Consider the stochastic reaction–diffusion equation on a bounded smooth domain $\mathcal D\subset\mathbb R^d$ with Dirichlet boundary conditions, obtained from (4.10) by replacing…

Let $\mathcal{E}(x,t)$ solve the diffraction-amplification equation on a $d$-dimensional torus $\Lambda\subset\mathbb{R}^d$, with diffusion constant $\mathcal{D}\geq 0$ replaced by…

Rarefaction stability conjecture. The long-time behavior of solutions is governed by the smoothed Riemann solution of the corresponding inviscid system.

Large-scale convergence conjecture. Under the scaling assumption on the noise, the expected density should satisfy

Compact support conjecture. The same compact-support and unbounded-union-of-supports results should hold for solutions to $$ .

Let method denote the global gradient-taming method considered in the paper, and let the spatial discretization be defined on a domain. Global gradient-taming suitability conjectur…

Consider the spectral Galerkin space discretization discussed in the paper, with projection $P_h$ onto the retained Fourier modes, and the numerical schemes tested for the SPDE. Th…

Consider the SPDE $du=(-u^3+Delta u)\,dt+\sqrt{2}\,dW_t$ and its semi-implicit Euler time discretization … Finite-time blowup conjecture. The above time discretization will exhibit…

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 a one-dimensional microscopic interface growth model belong to a broad class of such models, and let $\mathcal H$ denote its macroscopic fluctuation field. The KPZ equation is…

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…

Let $d\geq 3$. For $c_1>0$, let $\mathcal{L}_0$ be the Ornstein–Uhlenbeck operator, let $\mathcal{A}_+^N$ and $\mathcal{A}_-^N$ be the components of the generator…