24 problems

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 $D\subset\mathbb R^d$ be open, bounded, and convex, and let $\mathcal W$ be a renormalization class on $\overline D$. Assume that the asymptotic fixed-point equation has a uniq…

KPZ fluctuation conjecture. At the critical scaling, one-dimensional diffusions in random media should exhibit fluctuations governed by the KPZ equation.

Let $M$ be a closed manifold of dimension $d$, let $V$ be the potential defining the invariant measure $mu$ for the diffusion generator, and let $p$ be the Wasserstein exponent. Fo…

Let $b1 b2$ and $d_m$ satisfy … and … Here $b1$ is the volume growth exponent, $b2$ is the escape time exponent, and $d_m$ is the martingale dimension, also called the martingale i…

Let a diffusion on a metric measure space have spectral dimension, and let its martingale dimension be defined as the dimension of its martingale representation. Hino proved that f…

A sub-Riemannian diffusion killed when it exits a sufficiently small metric ball is expected to exhibit an analogous small-ball spectral convergence to the corresponding nilpotent…

The two-parameter processes $\operatorname{PDIPE}(\alpha,\theta)$ are diffusions on the boundary of the branching graph of integer compositions. An up-down chain on this graph adds…

Let $\mathcal{A}$ be a manifold in $\mathbb{R}^d$ without boundary, and let $(X_t,\eta_t)$ be a diffusion process on $\mathbb{R}\times\mathcal{A}$. Assume that $\eta_t$ has station…

Zero-noise projection conjecture. In this limit, the model recovers projections to the latent space in a sense analogous to ordinary probabilistic principal component analysis, whe…

Local spectral dimension conjecture. The local spectral dimension for the non-fixed-point diffusion should be the spectral dimension for the limiting fractal in general.

The Kolmogorov diffusion is a smooth non-trivial nilpotent diffusion, and a Markovian maximal coupling is a Markovian coupling that achieves the maximal possible coupling probabili…

Let $X_{z,z',w,w'}$ be the Markov process on the space $Omega$ constructed from the quadruple $(z,z',w,w')$, and let its generator be implemented by the differential operator…

Analytic-coefficient moment parametrization conjecture. This parametrization holds when $a(x,\boldsymbol\theta)$ and $b(x,\boldsymbol\theta)$ are analytic in $x$ and…

Analytic moment parametrization conjecture. If $a(x,\boldsymbol\theta)$ and $b(x,\boldsymbol\theta)$ are analytic in $x$ and $\boldsymbol\theta$, then $\boldsymbol\theta$ is a gene…

Self-reinforced diffusion conjecture. If $\sum_{k=1}^{n}k^2a_k<0$, then $X_t$ converges almost surely; if $\sum_{k=1}^{n}k^2a_k>0$, then

Let $\delta_1=\frac 12-\frac d{2r}-\frac 1q$ be the parameter governing the stated moment and regularity bounds for the stochastic flow associated with the diffusion and drift unde…

Quantum characteristic-function conjecture. In the high-energy, semiclassical scaling, for the relevant parameters $t$ and $q$,

Let $\pi(w)$ be the target density, constrained to the interval $[\ell^{\downarrow}_{s,t},\ell^{\uparrow}_{s,t}]$, and let $\upsilon^{\downarrow}_{s,t}$ be the associated endpoint…

Large-drift boundedness conjecture. For a deterministic jump distribution $\nu=\delta_{x_0}$, the spectral gap remains bounded as $\mu\to\infty$, whereas $\lambda_0^D$ tends to inf…

Threshold conjecture. The threshold is

Let $\tau_0$ be the first passage time to $0$, let $\lambda_x(t)$ be its rate function, and let $\mu^n_1$ be the principal eigenvalue of the Dirichlet problem on $(0,n)$ associated…

Multidimensional martingale diffusion limit conjecture. For $d>1$, any $d$-dimensional cadlag stochastic process is a limit of martingale diffusions in the sense of finite-dimensio…