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Lucian Beznea

Publications and source records attributed to Lucian Beznea.

At least 19 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↗
arXiv2024

Continuous flows driving Markov processes and multiplicative $L^p$-semigroups

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 order operator to the generator of a Markov process, when the two generators commute. A relevant example is a measure-valued superprocess having a continuous flow as spatial motion and a branching mechanism which does not depend on the spatial variable. We prove that any flow is actually continuous in a convenient topology and we show that a Markovian multiplicative semigroup on an Lp space is generated by a continuous flow, completing the answer to the question whether it is enough to have a measurable structure, like a C0-semigroup of Markovian contractions on an $L^p$-space with no fixed topology, in order to ensure the existence of a right Markov process associated to the given semigroup. We extend from bounded to unbounded functions the weak generator (in the sense of Dynkin) and the corresponding martingale problem

math.PR↗
arXiv2024

Nonlinear Dirichlet problem of non-local branching processes

We present a method of solving a nonlinear Dirichlet problem with discontinuous boundary data and we give a probabilistic representation of the solution using the nonlocal branching process associated with the nonlinear term of the operator. Instead of the pointwise convergence of the solution to the given boundary data we use the controlled convergence which allows to have discontinuities at the boundary.

math.PR↗
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↗
arXiv2023

Nonlinear Dirichlet forms associated with quasiregular mappings

If $({\cal E}, {\cal D})$ is a symmetric, regular, strongly local Dirichlet form on $L^2 (X,m)$, admitting a carré du champ operator $Γ$, and $p>1$ is a real number, then one can define a nonlinear form ${\cal E}^p$ by the formula $$ {\cal E}^p(u,v) = \int_{X} Γ(u)^\frac{p-2}{2} Γ(u,v)dm , $$ where $u$, $v$ belong to an appropriate subspace of the domain ${\cal D}$. We show that ${\cal E}^p$ is a nonlinear Dirichlet form in the sense introduced by P. van Beusekom. We then construct the associated Choquet capacity. As a particular case we obtain the nonlinear form associated with the $p$-Laplace operator on $W_0^{1,p}$. Using the above procedure, for each $n$-dimensional quasiregular mapping $f$ we construct a nonlinear Dirichlet form ${\cal E}^n$ ($p=n$) such that the components of $f$ become harmonic functions with respect to ${\cal E}^n$. Finally, we obtain Caccioppoli type inequalities in the intrinsic metric induced by ${\cal E}$, for harmonic functions with respect to the form ${\cal E}^p$.

math.AP↗
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

From Monte Carlo to neural networks approximations of boundary value problems

In this paper we study probabilistic and neural network approximations for solutions to Poisson equation subject to Holder data in general bounded domains of $\mathbb{R}^d$. We aim at two fundamental goals. The first, and the most important, we show that the solution to Poisson equation can be numerically approximated in the sup-norm by Monte Carlo methods, and that this can be done highly efficiently if we use a modified version of the walk on spheres algorithm as an acceleration method. This provides estimates which are efficient with respect to the prescribed approximation error and with polynomial complexity in the dimension and the reciprocal of the error. A crucial feature is that the overall number of samples does not not depend on the point at which the approximation is performed. As a second goal, we show that the obtained Monte Carlo solver renders in a constructive way ReLU deep neural network (DNN) solutions to Poisson problem, whose sizes depend at most polynomialy in the dimension $d$ and in the desired error. In fact we show that the random DNN provides with high probability a small approximation error and low polynomial complexity in the dimension.

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

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↗
arXiv2018

Measure-valued branching processes associated with Neumann nonlinear semiflows

We construct a measure-valued branching Markov process associated with a nonlinear boundary value problem, where the boundary condition has a nonlinear pseudo monotone branching mechanism term $-β$, which includes as a limit case $β(u) = - u^{m}$, with $0 < m < 1$. The process is then used in the probabilistic representation of the solution of the parabolic problem associated with a nonlinear Neumann boundary value problem. In this way the classical association of the superprocesses to the Dirichlet boundary value problems also holds for the nonlinear Neumann boundary value problems. It turns out that the obtained branching process behaves on the measures carried by the given open set like the linear continuous semiflow, induced by the reflected Brownian motion, while the branching occurs on the measures having non-zero traces on the boundary of the open set, with the behavior of the $(-β)$-superprocess, having as spatial motion the process on the boundary associated to the reflected Brownian motion

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

Stochastic equation of fragmentation and branching processes related to avalanches

We give a stochastic model for the fragmentation phase of a snow avalanche. We construct a fragmentation-branching process related to the avalanches, on the set of all fragmentation sizes introduced by J. Bertoin. A fractal property of this process is emphasized. We also establish a specific stochastic equation of fragmentation. It turns out that specific branching Markov processes on finite configurations of particles with sizes bigger than a strictly positive threshold are convenient for describing the continuous time evolution of the number of the resulting fragments. The results are obtained by combining analytic and probabilistic potential theoretical tools.

math.PR↗
arXiv2015

Measure-valued discrete branching Markov processes

We construct and study branching Markov processes on the space of finite configurations of the state space of a given standard process, controlled by a branching kernel and a killing one. In particular, we may start with a superprocess, obtaining a branching process with state space the finite configurations of positive finite measures on a topological space. A main tool in proving the path regularity of the branching process is the existence of convenient superharmonic functions having compact level sets, allowing the use of appropriate potential theoretical methods.

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↗
arXiv2011

On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. A second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels .

math.PR↗