arXiv subjects

Iulian Cîmpean

Publications and source records attributed to Iulian Cîmpean.

15 recordsLinked to original sources

arXiv2026

Regularization of the superposition principle: Potential theory meets Fokker-Planck equations

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 marginals equal to this solution, and additionally solving the martingale problem for the Kolmogorov operator given by the FPE. The superposition principle thus reveals that such parabolic PDEs have a probabilistic counter part. The aim of this work is to go a substantial further step and, by exploiting the superposition principle, construct a full fledged Markov process, i.e. a family of path space measures for a large set of space time starting points connected by the Markov property, associated to the (linearized) FPE in the above way. Under very general (merely measurability) conditions on the coefficients of the FPE this is achieved in this paper in such a way that the resulting process is a right process, which is a particularly useful class of Markov processes, enjoying among other regularity properties the strong Markov property, which is fundamental for the analysis of the underlying FPE as a (nonlinear) parabolic PDE by probabilistic tools. As two main applications we construct fundamental flow solutions for the FPE and we prove a well-posedeness result for the parabolic Dirichlet problem through probabilistic means for more general coefficients than could be treated in the existing literature. Furthermore, we introduce a Choquet capacity for such FPEs using the corresponding right process. The validity of the strong Markov property in the context of the superposition principle was an open problem even in the linear case. In this paper we solve this also in the nonlinear case, i.e. for path laws of solutions to McKean-Vlasov SDEs with Nemytskii type coefficients. A main application here is the FPE given by the generalized porous media equation and its corresponding McKean-Vlasov SDE.

math.PR↗

arXiv2026

Extended Walk-on-Spheres Algorithm for Linear and Nonlinear Elliptic Problems of Divergence-type

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 equation with Dirichlet boundary conditions. Its mesh-free nature, robustness on complex geometries, favorable scaling with dimension, and intrinsic parallelism distinguish it from mesh-based solvers. However, its efficient applicability has been essentially limited to operators that admit explicit probabilistic exit laws, excluding most variable-coefficient and nonlinear elliptic operators. We propose a general framework that aims to overcome this limitation by using the classical Dirichlet Laplacian and harmonic extension as universal building blocks. Rather than seeking a custom stochastic representation for each operator, we employ Walk-on-Spheres to precompute a reusable numerical operator toolbox that approximates the inverse Dirichlet Laplacian, the harmonic extension operator, and their gradients. These precomputed operators are then used to represent candidate solutions and to transform arbitrary Dirichlet boundary value problems into a finite-dimensional algebraic system/optimization problem for an unknown source term. Solving the resulting algebraic system/optimization problem and substituting back yields an approximate solution to the original PDE. Even more, for a general linear second order elliptic operator, the above mentioned precomputed toolbox can be directly used to obtain not just an approximation of a certain solution of the corresponding generalized Dirichlet problem, but an estimator of both the Green's integral operator and the elliptic measure operator. Numerical experiments on a range of benchmarks, including non-symmetric and anisotropic linear elliptic equations, semilinear and quasilinear problems, demonstrate the method's flexibility and efficiency.

math.NA↗

arXiv2025

Error analysis for the deep Kolmogorov method

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 work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.

math.NA↗

arXiv2025

Quantitative boundary Hölder estimates for the inhomogeneous Poisson problem through a probabilistic approach

In this paper we derive quantitative boundary Hölder estimates, with explicit constants, for the inhomogeneous Poisson problem in a bounded open set $D\subset \mathbb{R}^d$. Our approach has two main steps: firstly, we consider an arbitrary $D$ as above and prove that the boundary $α$-Hölder regularity of the solution the Poisson equation is controlled, with explicit constants, by the Hölder seminorm of the boundary data, the $L^ γ$-norm of the forcing term with $γ>d/2$, and the $α/2$-moment of the exit time from $D$ of the Brownian motion. Secondly, we derive explicit estimates for the $α/2$-moment of the exit time in terms of the distance to the boundary, the regularity of the domain $D$, and $α$. Using this approach, we derive explicit estimates for the same problem in domains satisfying exterior ball conditions, respectively exterior cone/wedge conditions, in terms of simple geometric features. As a consequence we also obtain explicit constants for pointwise estimates for the Green function and for the gradient of the solution. The obtained estimates can be employed to bypass the curse of high dimensions when aiming to approximate the solution of the Poisson problem using neural networks, obtaining polynomial scaling with dimension, which in some cases can be shown to be optimal.

math.PR↗

arXiv2025

A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis

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 problems aim at solving an elliptic partial differential equation (PDE) or a system of elliptic PDEs in a bounded Euclidean domain, subject to incomplete boundary and/or internal conditions, and are usually severely ill-posed. In a very recent paper \cite{CiGrMaI}, a probabilistic numerical framework has been developed by the authors, wherein such inverse problems could be analysed thoroughly by simulating the spectrum of some corresponding direct problem and its singular value decomposition based on stochastic representations and Monte Carlo simulations. Herein a full probabilistic error analysis of the aforementioned methods is provided, whereas the convergence of the corresponding approximations is proved and explicit error bounds are provided. This is achieved by employing tools from several areas such as spectral theory, regularity theory for elliptic measures, stochastic representations, and concentration inequalities.

math.NA↗

arXiv2025

Numerical spectral analysis of Cauchy-type inverse problems: A probabilistic approach

We investigate the inverse Cauchy and data completion problems for elliptic partial differential equations in a bounded domain $D \subset \mathbb{R}^d$, $d \ge 2$, with a special emphasis on the steady-state heat conduction in anisotropic media. More precisely, boundary conditions are prescribed on an accessible part of the boundary $\varnothing \neq Γ_0 \subsetneqq \partial{D}$ and/or internal conditions are available inside the domain $D$ and the aim is to reconstruct the solution to these inverse problems in the domain and on the inaccessible remaining boundary $Γ_1 := \partial{D} \setminus Γ_0$. Although such severely ill-posed problems have been studied intensively in the past decades, deriving efficient methods for approximating their solution still remains challenging in the general setting, e.g., in high dimensions, for solutions and/or domains with singularities, in complex geometries, etc. Herein, we derive a fundamental probabilistic framework for the stable reconstruction of the solution to the Cauchy and data completion problems in steady-state anisotropic heat conduction, as well as enhancing the knowledge on the impact of the geometry of the domain $D$ and the structure of the conductivity tensor $\mathbf{K}$ on the stability of these inverse problems. This is achieved in three steps: ({\it i}) the spectrum of the direct problem is simulated using stochastic estimators; ({\it ii}) the singular value decomposition of the corresponding direct operator is performed; and ({\it iii}) for the prescribed measurements, a natural subspace of approximate solutions is constructed. This approach is based on elliptic measures, in conjunction with probabilistic representations and parallel Monte Carlo simulations. Thorough numerical simulations performed on GPU, for various two- and three-dimensional geometries, are also provided.

math.NA↗

arXiv2024

Construction of Hunt processes by the Lyapunov method and applications to generalized Mehler semigroups

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 remained open for more than a decade. Our approach is to reconsider the {\it càdlàg problem} for generalized Mehler semigroups as a particular case of the much broader problem of constructing Hunt processes from a given Markov semigroup. Following this strategy, a consistent part of this work is devoted to prove that starting from a Markov semigroup on a general (possibly non-metrizable) state space, the existence of a suitable Lyapunov function with relatively compact sub/sup-sets in conjunction with a local Feller-type regularity of the resolvent are sufficient to ensure the existence of an associated càdlàg Markov process; if the topology is locally generated by potentials, then the process is in fact Hunt. Other results of fine potential theoretic nature are also pointed out, an important one being the fact that the Hunt property of a process is stable under the change of the topology, as long as it is locally generated by potentials. Then, we derive sufficient conditions for a large class of generalized Mehler semigroups in order to posses an associated Hunt process with values in the original space. To this end, we first construct explicit Lyapunov functions whose sub-level sets are relatively compact with respect to the (non-metrizable) weak topology, and then we use the above mentioned stability to deduce the Hunt property with respect to the stronger norm topology. We test these conditions on a stochastic heat equation on $L^2(D)$ whose drift is the Dirichlet Laplacian on a bounded domain $D \subset \mathbb{R}^d$, driven by a (non-diagonal) Lévy noise whose characteristic exponent is not necessarily Sazonov continuous.

math.PR↗

arXiv2022

Strong Feller semigroups and Markov processes: A counter example

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 Markov kernels on a Polish space are not enough to ensure the existence of an associated càdlàg Markov process on the same space. One such simple counter-example is the Brownian semigroup on $\mathbb{R}$ restricted to $\mathbb{R}\setminus \{0\}$, for which it is shown that there is no associated càdlàg Markov process. Using the same idea and results from potential theory we then prove that the analogous result with càdlàg Markov process replaced by right Markov process also holds, even if one allows to change the Polish topology to another Polish topology with the same Borel $σ$-algebra.

math.PR↗

arXiv2022

Path continuity of Markov processes and locality of Kolmogorov operators

We prove that if we are given a generator of a cadlag Markov process and an open domain $G$ in the state space, on which the generator has the local property expressed in a suitable way on a class $\mathcal{C}$ of test functions that is sufficiently rich, then the Markov process has continuous paths when it passes through $G$. The result holds for any Markov process which is associated with the generator merely on $\mathcal{C}$. This points out that the path continuity of the process is an a priori property encrypted by the generator acting on enough test functions, and this property can be easily checked in many situations. The approach uses potential theoretic tools and covers Markov processes associated with (possibly time-dependent) second order integro-differential operators (e.g., through the martingale problem) defined on domains in Hilbert spaces or on spaces of measures.

math.PR↗

arXiv2019

The nonlinear Schrödinger equation with white noise dispersion on quantum graphs

We show that the nonlinear Schrödinger equation (NLSE) with white noise dispersion on quantum graphs is globally well-posed in $L^2$ once the free deterministic Schrödinger group satisfies a natural $L^1-L^{\infty}$ decay, which is verified in many examples. Also, we investigate the well-posedness in the energy domain in general and in concrete situations, as well as the fact that the solution with white noise dispersion is the scaling limit of the solution to the NLSE with random dispersion.

math.AP↗

arXiv2019

A natural extension of Markov processes and applications to singular SDEs

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 standard trivial extension in which case the process is made stuck in the added points, and it renders a new technique of constructing extended solutions to S(P)DEs from all starting points, in such a way that they are solutions at least after any strictly positive time. Concretely, we adopt this strategy to study SDEs with singular coefficients on an infinite dimensional state space (e.g. SPDEs of evolutionary type), for which one often encounters the situation where not every point in the space is allowed as an initial condition. The same can happen when constructing solutions of martingale problems or Markov processes from (generalized) Dirichlet forms, to which our new technique also applies.

math.PR↗

arXiv2017

Invariant, super and quasi-martingale functions of a Markov process

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 the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-excessive functions, we present a two-step approach to the existence of invariant probability measures.

math.PR↗

arXiv2017

Quasimartingales associated to Markov processes

For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the dif- ference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.

math.PR↗

arXiv2016

A new approach to the existence of invariant measures for Markovian semigroups

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 prove conditions equivalent to the existence of an invariant finite measure which is absolutely continuous with respect to it. As applications, we give a short proof for the result of Lasota and Szarek on invariant measures and we obtain a unifying generalization of different versions for Harris ergodic theorem which provides an answer to an open question of Tweedie. We show that for a nonlinear SPDE on a Gelfand triple, the strict coercivity condition is sufficient to guarantee the existence of a unique invariant probability measure for the associated semigroup, once it satisfies a Harnack type inequality. A corollary of the main result shows that any uniformly bounded semigroup on $L^p$ possesses an invariant measures and we give some applications to sectorial perturbations of Dirichlet forms.

math.PR↗

arXiv2015

Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents

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 $m$. We give a unifying characterization of the invariant functions, revealing the fact that an $L^p$-integrable function is harmonic if and only if it is harmonic with respect to the weak dual resolvent. Our approach is based on potential theoretical techniques for resolvents in weak duality. We prove the equivalence between the $m$-irreducible recurrence of the resolvent and the extremality of $m$ in the set of all invariant measures, and we apply this result to the extremality of Gibbs states. We also show that our results can be applied to non-symmetric Dirichlet forms, in general and in concrete situations. A second application is the extension of the so called Fukushima ergodic theorem for symmetric Dirichlet forms to the case of sub-Markovian resolvents of kernels.

math.PR↗