32 problems

Let $n\geq 1$, let $\mu_n$ be a mean-zero Gaussian measure on $\mathbb{R}^n$, and let $K,M\subseteq\mathbb{R}^n$ be closed convex sets symmetric about the origin. Gaussian correlat…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and define the centered Gaussian probability measure … where…

Let $f$ be a function on $(0,1)$, let $V$ be a symmetric convex set in $\mathbb{R}^n$, and let the relevant lattice-covering parameter and Gaussian measure be as defined in the sur…

Let $c(F)$ be a separable Banach space and let $cmu$ be a centered Gaussian measure on $F$. Gaussian correlation conjecture. [The supplied candidate is incomplete and contains no a…

Let $\gamma$ denote the standard Gaussian measure on $\mathbb R$. For $n,d\in\mathbb N$, let $\mathcal P_d(\mathbb R^n)$ be the space of polynomials on $\mathbb R^n$ of degree at m…

Let $d$ be a positive integer, let $(\mathbb{R}^d,g,\mu)$ satisfy the curvature-dimension condition $\operatorname{CD}(\rho,\infty)$ for some $\rho>0$, and let…

Let $\gamma$ denote the standard Gaussian density on $\mathbb{R}^n$, and let … where $p\in[0,1]$ and $a,b\in\mathbb{R}^n$. For a measurable set $E\subset\mathbb{R}^n$, consider the…

Let $\boldsymbol{A},\boldsymbol{B}\rhd 0$ be positive-definite covariance operators on $\mathcal{H}=\mathbb{R}^{d}$, and let $\pi_{\varepsilon}$ and $\pi_{0}$ denote the entropic a…

Let $a$ denote the fixed parameter from the definition of $A$, let $b>0$, and let $\gamma_n[b]$ denote the Gaussian measure or distribution appearing in $A(a,\gamma_n[b])$. The qua…

Consider the non-Gaussian component of the posterior measure arising in the GP-PDE collocation method, where collocation points impose nonlinear PDE observations. Dense-collocation…

Let $\eta^\mathbf y_\beta$ denote the finite-dimensional non-Gaussian measure arising in the decomposition of the conditioned Gaussian measure, with observation data $\mathbf y$ an…

Gaussian $n^{-1}$ convergence conjecture.

Optimal growth conjecture. There exists $C(s)>0$ such that, for every $t\in[0,t_0)$ and every $s\in(0,\sigma]$,

Positive-growth conjecture. There is no sequence $(a_n)$ in $cmathcal{M}^{\sigma}$ for which $b3_{s,(a_n)}<0$, and data randomly initialized in this manner grow in $H^s$ on average…

Exceptional-support conjecture. Part (i) of the theorem on invariant measures already contains essentially all counterexamples to non-invariance: for…

Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$, and let $\mu$ be a log-Lipschitz perturbation of $\gamma$. Let $T=\nabla\varphi$ be the Brenier map transporting…

Dashti–Stuart conjecture. Assume that the conditions in Assumption $4$ hold. Then:

Let $\nu$ be a Gaussian distribution, and let $\phi$ be the convex function constructed in the paper's fixed-point theorem. In the Gaussian case, transport maps between Gaussian di…

Let $\mu$ and $\nu$ be Gaussian measures on possibly different Euclidean spaces, and consider the Gromov–Wasserstein problem with squared ground distance. The Gaussian-restricted p…

Let $\mu=\mathcal{N}(0,\operatorname{diag}(\alpha))$ and $\nu=\mathcal{N}(0,\operatorname{diag}(\beta))$ be Gaussian measures, and let $GGW_2(\mu,\nu)$ denote the Gromov–Wasserstei…

Failure conjecture. A version of Theorem SimplyMain does not hold for the Gaussian measure generated by this Ornstein–Uhlenbeck process.

U-conjecture. There exist an orthogonal transformation $L$ of $\mathbb{R}^n$ and an integer $k$ such that $f\circ L$ is a function of $(x_1,\ldots,x_k)$ and $g\circ L$ is a functio…

Let $d$ be the dimension and let $\phi$ satisfy the normalization, regularity, and weighted Poincaré assumptions denoted by … – … int{mathbb R^d} xi xj e^{-phi(x)},mathrm d x=delta…

Let $K$ be a local field with residue-field cardinality $q$, let $L$ be a lattice in $K^d$, and let $V$ be the tropicalization of a full-dimensional Gaussian in $K^d$ with lattice…

Let $B_0(r)$ be the ball of radius $r$ centered at the origin, and let $S$ range over simplices containing this ball. Let the Gaussian measure have density proportional to…