25 problems

Let $B(0,1)\subset\mathbb{R}^d$ be the unit ball. Let $\lambda\in C^\infty(\overline{B(0,1)})$ satisfy $\lambda>0$ on $\overline{B(0,1)}$, and let…

Consider the diffusion operator on $[0,infty)$ … and define … The uniqueness criterion is $\int_0^\infty\mu[0,x]e^{-C(x)}=\infty$. Strong ergodicity criterion. The criterion for st…

Consider a linear time-varying (LTV) forward process and its reverse diffusion. An input-affine reverse diffusion is one whose dependence on the reverse-process input is affine. In…

Let $D\sigma\in L_{1,\operatorname{loc}}(\mathbb{R}^{d+1})$ and write $b=b_M+b_B$, where both summands are Borel measurable, $b_M$ is the Morrey part, and $b_B$ is the bounded part…

The paper studies discrete-time Bellman equations for policy evaluation in continuous-time diffusion processes, with theoretical guarantees proved under global boundedness and Lips…

The paper considers continuous-time diffusion processes whose diffusion coefficient may be degenerate, in contrast to the uniformly elliptic setting used for the preceding approxim…

Let $(M,\rho)$ be a connected complete compact weighted Riemannian manifold without boundary, equipped with a weighted probability measure $\mathrm{d}\mu=\mathrm{e}^V\mathrm{d}x$,…

Existence and Itô-formula conjecture. The process $\big((x(\cdot),i(\cdot)),\ell(\cdot)\big)$ exists and satisfies

Non-compact extension conjecture. Theorem ${\rm{}}$ should still hold when $M$ is non-compact after imposing the additional conditions described above.

Pure-jump limiting semigroup conjecture. In general, the limiting semigroup $e^{t\mathcal{L}_\infty}$ will be associated with a pure-jump continuous-time Markov process living in t…

Let $Y_t$ be the $Z$-measure diffusion with parameters $z,z',\vartheta$, and let $n$ denote the size of a partition state. A partition-valued jump process is a Markov process whose…

Let $V:\mathbb{R}^d\to\mathbb{R}$ be an even potential, $V(x)=V(-x)$, and consider the associated one-particle generator with eigenfunctions $(e_i)$ ordered as in the source. Highe…

The system consists of particles whose branching law has heavy tails and whose motion is governed by a strongly mixing diffusion process. The Ornstein–Uhlenbeck process is a partic…

Let $G$ be a graph whose vertices are initially labelled by integers, interpreted as chips, and at each step let every vertex simultaneously fire one chip to each neighbouring vert…

Let $\eta_n$ be the system-size scaling parameter and let $d$ be the number of queues sampled by each arriving customer. Consider queues whose lengths are of order … The process tr…

General absorbed-diffusion conjecture. The equality

The stochastic diffusive behavior conjecture. For a generic perturbation $\varepsilon H_1(\cdot)$, there are smooth functions $b(I)$ and $\sigma(I)>0$, depending only on $H_1$ and…

Diffusion conjecture. The limit process $\chi$ is a diffusion with $\mathscr{L}_t$ as its generator.

Averaged-generator conjecture. Comparing the variance derivative above with $\frac{dK(t,t)}{dt}$, we conjecture that $\tilde W$ is a Gaussian diffusion process with infinitesimal g…

Classical-solution conjecture. The abstract Cauchy problem has a unique classical solution in the sense of Definition II.6.1 of Ethier and Kurtz: the mapping $t\mapsto u(t,\cdot)$…

Let $R^{(k)}$ be the $(0,\infty)^k$-valued weak solution of the limiting diffusion, with $k$ variable, defined by … where $B_1,\ldots,B_{k+1}$ are independent one-dimensional stand…

Stronger converse conjecture. Then $\pi$ is a nonnegative measure and consequently it is a stationary distribution of $X$.

Gaussian-tail conjecture. The limit queue length process has a Gaussian tail, and the tail depends on the service time distribution only through its first two moments.

A diffusion process is a continuous strong Markov process, and its natural filtration is the filtration generated by the process. A filtration is continuous when every martingale o…

Let ${\mathbf F}=\{{\mathcal F}_t\}_{t\in[0,\infty)}$ be the filtration induced by the canonical diffusion process on a nested fractal. An AF-martingale dimension of one means that…