Related papers: Regularity of harmonic functions for a class of si…

We obtain sharp estimates for functions harmonic with respect to $x$-dependent rectilinear stable processes in balls, under the assumption that the Dirichlet exterior data are radial about the center. The main idea of the proof is based on…

Martin boundaries and integral representations of positive functions which are harmonic in a bounded domain $D$ with respect to Brownian motion are well understood. Unlike the Brownian case, there are two different kinds of harmonicity with…

We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In…

We consider nonlinear integro-differential equations, like the ones that arise from stochastic control problems with purely jump L\`evy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior…

We consider some classes of Levy processes for which the estimate of Krylov and Safonov (as in [BL02]) fails and thus it is not possible to use the standard iteration technique to obtain a-priori Holder continuity estimates of harmonic…

We consider systems of stochastic differential equations of the form \[ \d X_t^i = \sum_{j=1}^d A_{ij}(X_{t-}) \d Z_t^j\] for $i=1,\dots,d$ with continuous, bounded and non-degenerate coefficients. Here $Z_t^1,\dots,Z_t^d$ are independent…

We prove the existence of solutions for the stochastic differential equation $dX_t=b(t,X_{t-})dZ_t+a(t,X_t)dt, X_0\in\R, t\ge 0,$ with only measurable coefficients $a$ and $b$ satisfying the condition $0<\mu\le |b(t,x)|\le \nu$ and…

We propose stochastic $N$-component synchronization models $(x_{1}(t),...,x_{N}(t))$, $x_{j}\in\mathbb{R}^{d}$, $t\in\mathbb{R}_{+}$, whose dynamics is described by Levy processes and synchronizing jumps. We prove that symmetric models…

In this paper we consider the existence of weakly c\`adl\`ag versions of a solution to a linear equation in a Hilbert space $H$, driven by a Levy process taking values in a Hilbert space $U$. In particular we are interested in diagonal type…

Let $X_1,...,X_N$ denote $N$ independent $d$-dimensional L\'evy processes, and consider the $N$-parameter random field \[\X(\bm{t}):= X_1(t_1)+...+X_N(t_N).\] First we demonstrate that for all nonrandom Borel sets $F\subseteq\R^d$, the…

Let u, v be two harmonic functions in the disk of radius two which have exactly the same set Z of zeros. We observe that the gradient of \log |u/v| is bounded in the unit disk by a constant which depends on Z only. In case Z is empty this…

New explicit exponential stability conditions are presented for the non-autonomous scalar linear functional differential equation $$ \dot{x}(t)+ \sum_{k=1}^m a_k(t)x(h_k(t))+\int_{g(t)}^t K(t,s) x(s)ds=0, $$ where $h_k(t)\leq t$, $g(t)\leq…

We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz '63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant…

We introduce new sufficient conditions for verifying stability and recurrence properties in singularly perturbed stochastic hybrid dynamical systems. Specifically, we focus on hybrid systems with deterministic continuous-time dynamics that…

We consider the Hammersley-Aldous-Diaconis (HAD) process with sinks and sources such that there is a microscopic shock at every time $t$; denote $Z(t)$ its position. We show that the mean and variance of $Z(t)$ are linear functions of $t$…

In this paper we estimate both the Hurst and the stable indices of a H-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at…

Exponential stability and solution estimates are investigated for a delay system $$ \dot{x}(t) - A(t)\dot{x}(g(t))=\sum_{k=1}^m B_k(t)x(h_k(t)) $$ of a neutral type, where $A$ and $B_k$ are $n\times n$ bounded matrix functions, and $g, h_k$…

In this paper, we provide the sufficient and necessary conditions for the symmetry of the following stable L\'evy-type operator $\mathcal{L}$ on $\mathbb{R}$: $$\mathcal{L}=a(x){\Delta^{\alpha/2}}+b(x)\frac{\d}{\d x},$$ where $a,b$ are the…

We consider the Harmonic crystal, a measure on $\mathbb{R}^{\mathbb{Z}^{d}}$ with Hamiltonian $H(\x)=\sum_{i,j}J_{i,j}(\x(i)-\x(j))^{2}+ h\sum_{i}(\x(i)-\dd(i))^{2}$, where $\x, \dd$ are configurations, $\x(i),\dd(i)\in\mathbb{R}$,…

Stochastic differential equations (SDEs) without global Lipschitz drift often demonstrate unusual phenomena. In this paper, we consider the following SDE on $\mathbb R^d$: \begin{align*} \mathrm{d} \mathbf{X}_t=\mathbf{b}(\mathbf{X}_t)…