Related papers: Feller generators with singular drifts in the crit…

We show the $L^r(\mathbb{R}^d, \mu)$-uniqueness for any $r \in (1, 2]$ and the essential self-adjointness of a Dirichlet operator $Lf = \Delta f +\langle \frac{1}{\rho}\nabla \rho , \nabla f \rangle$, $f \in C_0^{\infty}(\mathbb{R}^d)$ with…

We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $\Omega\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_\Omega…

Let $c_{kl} \in W^{2,\infty}(\mathbb{R}^d, \mathbb{C})$ for all $k,l \in \{1, \ldots, d\}$. We consider the divergence form operator $A = - \sum_{k,l=1}^d \partial_l (c_{kl} \, \partial_k) $in $L_2(\mathbb{R}^d)$ when the coefficient matrix…

We introduce a technique to obtain pointwise upper and lower bounds for the Green's function of elliptic operators whose principal part is the Laplacian and that include a drift term diverging near the boundary like a power of the inverse…

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(\alpha)}_\kappa u+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \…

Given a bounded domain in the Euclidean space satisfying the uniform outer cone condition, we show that a uniformly elliptic operator of second order with continuous second order coefficients generates a holomorphic semigroup on the space…

We consider percolation on the discrete torus $\mathbb{Z}_n^d$ at $p_c(\mathbb{Z}^d)$, the critical value for percolation on the corresponding infinite lattice $\mathbb{Z}^d$, and within the scaling window around it. We assume that $d$ is a…

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb…

We consider the energy supercritical defocusing nonlinear Schr\"odinger equation $i\partial_tu+\Delta u-u|u|^{p-1}=0$ in dimension $d\ge 5$. In a suitable range of energy supercritical parameters $(d,p)$, we prove the existence of $\mathcal…

In this paper we establish a rigidity property of holomorphic generators by using their local behavior at a boundary point $\tau$ of the open unit disk $\Delta$. Namely, if $f\in\mathrm{Hol}(\Delta,\mathbb{C})$ is the generator of a…

We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection-diffusion equation \[ \partial_t \theta - \Delta \theta + b \cdot \nabla \theta = 0 \] to satisfy local boundedness, a single-scale…

By using lower bound conditions of the L\'evy measure w.r.t. a nice reference measure, the coupling and strong Feller properties are investigated for the Markov semigroup associated with a class of linear SDEs driven by (non-cylindrical)…

We consider second order differential operators $A_\mu$ on a bounded, Dirichlet regular set $\Omega \subset \mathbb{R}^d$, subject to the nonlocal boundary conditions \[ u(z) = \int_\Omega u(x)\, \mu (z, dx)\quad \mbox{for } z \in \partial…

We study the Dirichlet problem for a second order linear elliptic equation in a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, $n \ge 3$, with the drift $\mathbf{b} $ belonging to the critical weak space $L^{n,\infty}(\Omega )$. We…

We investigate a second order elliptic differential operator $A_{\beta, \mu}$ on a bounded, open set $\Omega\subset\mathbb{R}^{d}$ with Lipschitz boundary subject to a nonlocal boundary condition of Robin type. More precisely we have $0\leq…

We investigate properties of Markov quasi-diffusion processes corresponding to elliptic operators $L=a^{ij}D_{ij}+b^{i}D_{i}$, acting on functions on $\mathbb{R}^{d}$, with measurable coefficients, bounded and uniformly elliptic $a$ and…

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the…

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if $-A$ generates a $C_0$-semigroup on a…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

The famous 1960s Lumer-Phillips Theorem states that a closed and densely defined operator $A\colon D(A)\subseteq X\rightarrow X$ on a Banach space $X$ generates a strongly continuous contraction semigroup if and only if $(A,D(A))$ is…