Related papers: Intrinsic scaling properties for nonlocal operator…
We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a…
We study nonlocal operators acting on functions in the Euclidean space. The operators under consideration generate anisotropic jump processes, e.g., a jump process that behaves like a stable process in each direction but with a different…
We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump L\`evy processes. If the index of the…
We study weak solutions to nonlocal equations governed by integrodifferential operators. Solutions are defined with the help of symmetric nonlocal bilinear forms. Throughout this work, our main emphasis is on operators with general,…
In this work, we investigate quantitative regularity estimates for degenerate parabolic partial differential equations, with a focus on Orlicz-type diffusive structures. Using a geometric tangential analysis tailored to these structures and…
A piecewise-deterministic Markov process is a stochastic process whose behavior is governed by an ordinary differential equation punctuated by random jumps occurring at random times. We focus on the nonparametric estimation problem of the…
We establish the comparison principle and the existence of solutions of the integro-differential equations with L{\'e}vy operators. The L{\'e}vy operators of our interest are infinitesmal generator of the jump processes which could be…
Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having…
We develop a monotone, two-scale discretization for a class of integrodifferential operators of order $2s$, $s \in (0,1)$. We apply it to develop numerical schemes, and derive pointwise convergence rates, for linear and obstacle problems…
We extend the classical Bernstein technique to the setting of integro-differential operators. As a consequence, we provide first and one-sided second derivative estimates for solutions to fractional equations, including some convex fully…
We introduce and systematically develop two classes of discrete integrable operators: those with $2\times 2$ matrix kernels and those possessing general differential kernels, thereby generalizing the discrete analogue previously studied. A…
We suggest to investigate certain non-standard (pseudo-)differential operators in order to construct and to study multi-parameter processes. Our approach will include "classical" multi-parameter Markov processes but will go eventually far…
H\"older estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2.
In this paper we establish optimal regularity estimates and smoothness of free boundaries for nonlocal obstacle problems governed by a very general class of integro-differential operators with possibly singular kernels. More precisely, in…
We study stochastic differential equations with jumps with no diffusion part. We provide some basic stochastic characterizations of solutions of the corresponding non-local partial differential equations and prove the Harnack inequality for…
The paper deals with homogenization problem for a non-local linear operator with a kernel of convolution type in a medium with a periodic structure. We consider the natural diffusive scaling of this operator and study the limit behaviour of…
We consider fully nonlinear elliptic integro-differential operators with kernels of variable orders, which generalize the integro-differential operators of the fractional Laplacian type in \cite{CS}. Since the order of differentiability of…
In this work, we consider the regularity property of stochastic convolutions for a class of abstract linear stochastic retarded functional differential equations with unbounded operator coefficients. We first establish some useful estimates…
We consider here convolution operators, in the Caputo sense, with non-singular kernels. We prove that the solutions to some integro-differential equations with such operators (acting on the space variable) coincide with the transition…
We study parabolic equations governed by integro-differential operators with nonlocal components in some directions and local components in the remaining directions. The setting contains the purely nonlocal, as well as the purely local…