18 papers
Extended Walk-on-Spheres Algorithm for Linear and Nonlinear Elliptic Problems of Divergence-type
Iulian Cîmpean, Andreea Grecu, Arghir Zarnescu
The Walk-on-Spheres algorithm, introduced by M. E. Muller in 1956, is a well known Monte Carlo method that leverages Brownian exit distributions from spheres to solve the Laplace e…
Regularization of the superposition principle: Potential theory meets Fokker-Planck equations
Lucian Beznea, Iulian Cîmpean, Michael Röckner
For a solution to a (possibly nonlinear) Fokker-Planck equation (FPE) the powerful superposition principle renders a probability measure on path space with one dimensional time mar…
Quartic variation of the solution to the semilinear stochastic heat equation: limit behavior and asymptotic independence with respect to the data
I Cîmpean, Yassine Nachit, Ciprian A Tudor
This work concerns the limit behavior of the quartic variation (i.e., the power variation of order four) with respect to the time variable of the solution to the semilinear stochas…
Quantitative boundary Hölder estimates for the inhomogeneous Poisson problem through a probabilistic approach
Iulian Cîmpean, Ionel Popescu, Arghir Zarnescu
In this paper we derive quantitative boundary Hölder estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set . Our ap…
A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis
Iulian Cîmpean, Andreea Grecu, Liviu Marin
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse p…
Error analysis for the deep Kolmogorov method
Iulian Cîmpean, Thang Do, Lukas Gonon +2
The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this w…
Continuous flows driving Markov processes and multiplicative -semigroups
Lucian Beznea, Mounir Bezzarga, Iulian Cimpean
We develop a method of driving a Markov processes through a continuous flow. In particular, at the level of the transition functions we investigate an approach of adding a first or…
Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups
Lucian Beznea, Iulian Cîmpean, Michael Röckner
In this paper we deal with the problem of characterizing those generalized Mehler semigroups that do correspond to càdlàg Markov processes, which is highly non-trivial and has rema…
Numerical spectral analysis of Cauchy-type inverse problems: A probabilistic approach
Iulian Cîmpean, Andreea Grecu, Liviu Marin
We investigate the inverse Cauchy and data completion problems for elliptic partial differential equations in a bounded domain , , with a special e…
Strong Feller semigroups and Markov processes: A counter example
Lucian Beznea, Iulian Cîmpean, Michael Röckner
The aim of this note is to show, by providing an elementary way to construct counter-examples, that the strong Feller and the joint (space-time) continuity for a semigroup of Marko…
From Monte Carlo to neural networks approximations of boundary value problems
Lucian Beznea, Iulian Cimpean, Oana Lupascu-Stamate +2
In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of . We aim…
Path continuity of Markov processes and locality of Kolmogorov operators
Lucian Beznea, Iulian Cîmpean, Michael Röckner
We prove that if we are given a generator of a cadlag Markov process and an open domain in the state space, on which the generator has the local property expressed in a suitabl…
The nonlinear Schrödinger equation with white noise dispersion on quantum graphs
Iulian Cîmpean, Andreea Grecu
We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in once the free deterministic Schrödinger group s…
A natural extension of Markov processes and applications to singular SDEs
Lucian Beznea, Iulian Cîmpean, Michael Röckner
We develop a general method for extending Markov processes to a larger state space such that the added points form a polar set. The so obtained extension is an improvement on the s…
Invariant, super and quasi-martingale functions of a Markov process
Iulian Cîmpean, Lucian Beznea
We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with…
Quasimartingales associated to Markov processes
Lucian Beznea, Iulian Cîmpean
For a fixed right process we investigate those functions for which is a quasimartingale. We prove that is a quasimartingale if and only if is the dif- fer…
A new approach to the existence of invariant measures for Markovian semigroups
Lucian Beznea, Iulian Cîmpean, Michael Röckner
We give a new, two-step approach to prove existence of finite invariant measures for a given Markovian semigroup. First, we identify a convenient auxiliary measure and then we prov…
Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents
Lucian Beznea, Iulian Cîmpean, Michael Röckner
We analyze the transience, recurrence, and irreducibility properties of general sub- Markovian resolvents of kernels and their duals, with respect to a fixed sub-invariant measure…