Searcharxiv⌕ Search

arXiv subjects

Matija Vidmar

Publications and source records attributed to Matija Vidmar.

At least 19 recordsLinked to original sources
arXiv2026

Lecture notes: Probability with Measure

Lecture notes as per the title. In the first part, the concepts of a measurable space, measurable maps between measurable spaces and that of a measure on a measurable space are introduced, after which the fundamentals of the theory of Lebesgue integration are developed: convergence theorems, product spaces and Tonelli-Fubini, indefinite integration and absolute continuity, L-spaces and integral inequalities. Everything is set up so that in the second part the fundamental concepts of probability (such as those of random elements and their laws, independence, conditioning) can be cast swiftly in the measure-theoretic setting. Some emphasis is placed on monotone class and Dynkin's lemma type arguments. Products of arbitrary families of probabilities and Kolmogorov's extension theorem are treated.

math.PR↗
arXiv2025

Positive Markov processes in Laplace duality

This article develops a general framework for Laplace duality between positive Markov processes in which the one-dimensional Laplace transform of one process can be represented through that of another. We show that a process admits a Laplace dual if and only if it satisfies a certain complete monotonicity condition. Moreover, we analyse how the conventions adopted for the values of $0 \cdot \infty$ and $\infty \cdot 0$ are reflected in the weak continuity/absorptivity properties of the processes in duality at the boundaries $0$ and $\infty$. A broad class of generators admitting Laplace duals is identified, and we provide sufficient conditions under which the associated martingale problems are well-posed with the solutions being in duality at the level of their semigroups. Laplace duality is shown to furnish a unifying structure for several generalizations of continuous-state branching processes, e.g. those with immigration or evolving in random environments. Along the way, a theorem originally due to Ethier and Kurtz -- connecting duality of generators to that of the associated semigroups -- is refined, and we provide a concise proof of the Courr\`ege form for the pointwise infinitesimal generator of a positive Markov process whose domain includes the exponential functions. The latter leads naturally to the notion of a Laplace symbol, which is a parsimonious encoding of the infinitesimal dynamics of the process.

math.PR↗
arXiv2025

Symmetric splitting of one-dimensional noises

A symmetric random walk $X$ whose jumps have diffuse law, looked at up to an independent geometric random time, splits at the minimum into two independent and identically distributed pieces. The same for the maximum. It is natural to ask, are there any other times adapted to $X$ exhibiting this "symmetric splitting"? It appears that the phenomenon is most conveniently couched in terms of (what may be called) the noise structure of $X$. At the level of generality of the latter, an equivalent set-theoretic condition for the symmetric splitting property is provided, leading to the observation that the answer to the elucidated question is to the affirmative. While we do not deal much with the obvious analog of the phenomenon in continuous time, the discrete findings do beg the question: does linear Brownian motion admit times of symmetric splitting other than the maxima and minima? This is left unresolved, but we do make some comments as to why it may be non-trivial/interesting.

math.PR↗
arXiv2024

Fock structure of complete Boolean algebras of type I factors and of unital factorizations

The factorizable vectors of a complete Boolean algebra of type I factors, acting on a separable Hilbert space, are shown to be total, resolving a conjecture of Araki and Woods. En route, the spectral theory of noise-type Boolean algebras of Tsirelson is cast in the noncommutative language of "factorizations with unit" for which a muti-layered characterization of being "of Fock type" is provided.

math.OA↗
arXiv2024

Extending the noise of splitting to its completion and stability of Brownian maxima

The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The $\sigma$-fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made.

math.PR↗
arXiv2023

Continuous-state branching processes with collisions: first passage times and duality

We introduce a class of one-dimensional positive Markov processes generalizing continuous-state branching processes (CBs), by taking into account a phenomenon of random collisions. Besides branching, characterized by a general mechanism $Ψ$, at a constant rate in time two particles are sampled uniformly in the population, collide and leave a mass of particles governed by a (sub)critical mechanism $Σ$. Such CB processes with collisions (CBCs) are shown to be the only Feller processes without negative jumps satisfying a Laplace duality relationship with one-dimensional diffusions on the half-line. This generalizes the duality observed for logistic CBs by Foucart. Via time-change, CBCs are also related to an auxiliary class of Markov processes, called CB processes with spectrally positive migration (CBMs), recently introduced by Vidmar. We find necessary and sufficient conditions for the boundaries $0$ or $\infty$ to be attracting and for a limiting distribution to exist. The Laplace transform of the latter is provided. Under the assumption that the CBC process does not explode, the Laplace transforms of the first passage times below arbitrary levels are represented with the help of the solution of a second-order differential equation, whose coefficients express in terms of the Lévy-Khintchine functions $Σ$ and $Ψ$. Sufficient conditions for non-explosion are given.

math.PR↗
arXiv2023

A potpourri of results in the general theory of stochastic noises

The objects under inspection, on a given probability space, are noise(-type) Boolean algebras -- distributive non-empty sublattices of the lattice of all complete sub-$σ$-fields, whose every element admits an independent complement. Special attention is given to the spectral decompositions of the algebras of operators generated by the conditional expectations of their members (acting on $\mathrm{L}^2$). Atoms of the spectra are identified in explicit terms. For a reverse filtration admitting an innovating sequence of equiprobable random signs, a discreteness property of the spectral measure of the associated noise Boolean algebra is shown to imply product-typeness. Noise projections on the spectral space are introduced, which correspond to restricting a noise to a part of its domain space. They appear to play a natural (albeit technical) rôle in the general analysis. In particular through them manifests itself the tensor structure of the spectral decomposition. The spectrum is precisely delineated in the classical case (i.e. when the noise Boolean algebra is complete), a kind-of standard "symmetric Fock space" form thereof is procured. The latter result leads to a new characterization of classicality and blackness involving "spectral independence".

math.PR↗
arXiv2022

Stationary local random countable sets over the Wiener noise

The times of Brownian local minima, maxima and their union are three distinct examples of local, stationary, dense, random countable sets associated with classical Wiener noise. Being local means, roughly, determined by the local behavior of the sample paths of the Brownian motion, and stationary means invariant relative to the Lévy shifts of the sample paths. We answer to the affirmative Tsirelson's question, whether or not there are any others, and develop some general theory for such sets. An extra ingredient to their structure, that of an honest indexation, leads to a splitting result that is akin to the Wiener-Hopf factorization of the Brownian motion at the minimum (or maximum) and has the latter as a special case. Sets admitting an honest indexation are moreover shown to have the property that no stopping time belongs to them with positive probability. They are also minimal: they do not have any non-empty proper local stationary subsets. Random sets, of the kind studied in this paper, honestly indexed or otherwise, give rise to nonclassical one-dimensional noises, generalizing the noise of splitting. Some properties of these noises and the inter-relations between them are investigated. In particular, subsets are connected to subnoises.

math.PR↗
arXiv2022

Complete monotonicity of time-changed Lévy processes at first passage

We consider the class of (possibly killed) spectrally positive Lévy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which satisfy the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms.

math.PR↗
arXiv2022

Some characterizations for Markov processes at first passage

Suppose $X$ is a Markov process on the real line (or some interval). Do the distributions of its first passage times downwards (fptd) determine its law? In this paper we treat some special cases of this question. We prove that if the fptd process has the law of a subordinator, then necessarily $X$ is a Lévy process with no negative jumps; specifying the law of the subordinator determines the law of $X$ uniquely. We further show that, likewise, the classes of continuous-state branching processes and of self-similar processes without negative jumps are also respectively characterised by a certain structure of their fptd distributions; and each member of these classes separately is determined uniquely by the precise family of its fptd laws. The road to these results is paved by (i) the identification of Markov processes without negative jumps in terms of the nature of their fptd laws, and (ii) some general results concerning the identification of the fptd distributions for such processes.

math.PR↗
arXiv2022

Continuous-state branching processes with spectrally positive migration

Continuous-state branching processes (CSBPs) with immigration (CBIs), stopped on hitting zero, are generalized by allowing the process governing immigration to be any Lévy process without negative jumps. Unlike the CBIs, these newly introduced processes do not appear to satisfy any natural affine property on the level of the Laplace transforms of the semigroups. Basic properties are noted. Explicit formulae (on neighborhoods of infinity) for the Laplace transforms of the first passage times downwards and of the explosion time are derived.

math.PR↗
arXiv2021

Exit problems for positive self-similar Markov processes with one-sided jumps

A systematic exposition of scale functions is given for positive self-similar Markov processes (pssMp) with one-sided jumps. The scale functions express as convolution series of the usual scale functions associated with spectrally one-sided Lévy processes that underly the pssMp through the Lamperti transform. This theory is then brought to bear on solving the spatio-temporal: (i) two-sided exit problem; (ii) joint first passage problem upwards for the the pssMp and its multiplicative drawdown (resp. drawup) in the spectrally negative (resp. positive) case.

math.PR↗
arXiv2021

Some harmonic functions for killed Markov branching processes with immigration and culling

For a continuous-time Bienaymé-Galton-Watson process, $X$, with immigration and culling, $0$ as an absorbing state, call $X^q$ the process that results from killing $X$ at rate $q\in (0,\infty)$, followed by stopping it on extinction or explosion. Then an explicit identification of the relevant harmonic functions of $X^q$ allows to determine the Laplace transforms (at argument $q$) of the first passage times downwards and of the explosion time for $X$. Strictly speaking, this is accomplished only when the killing rate $q$ is sufficiently large (but always when the branching mechanism is not supercritical or if there is no culling). In particular, taking the limit $q\downarrow 0$ (whenever possible) yields the passage downwards and explosion probabilities for $X$. A number of other consequences of these results are presented.

math.PR↗
arXiv2021

A nonclassical solution to a classical SDE and a converse to Kolmogorov's zero-one law

For a discrete-negative-time discrete-space SDE, which admits no strong solution in the classical sense, a weak solution is constructed that is a (necessarily nonmeasurable) non-anticipative function of the driving i.i.d. noise. The result highlights the strong rôle measurability plays in (non-discrete) probability. En route one -- quite literally -- stumbles upon a converse to the celebrated Kolmogorov's zero-one law for sequences with independent values.

math.PR↗
arXiv2020

Double hypergeometric Lévy processes and self-similarity

Motivated by a recent paper of Budd, where a new family of positive self-similar Markov processes associated to stable processes appears, we introduce a new family of Lévy processes, called the double hypergeometric class, whose Wiener-Hopf factorisation is explicit, and as a result many functionals can be determined in closed form.

math.PR↗
arXiv2020

On laws exhibiting universal ordering under stochastic restart

For each of (i) arbitrary stochastic reset, (ii) deterministic reset with arbitrary period, (iii) reset at arbitrary constant rate, and then in the sense of either (a) first-order stochastic dominance or (b) expectation (i.e. for each of the six possible combinations of the preceding), those laws of random times are precisely characterized that are rendered no bigger [rendered no smaller; left invariant] by all possible restart laws (within the classes (i), (ii), (iii), as the case may be). Partial results in the same vein for reset with branching are obtained. In particular it is found that deterministic and arbitrary stochastic restart lead to the same characterizations, but this equivalence fails to persist for exponential (constant-rate) reset.

math.PR↗
arXiv2019

The structure of non-linear martingale optimal transport problems

We explore the structure of solutions to a family of non-linear martingale optimal transport (MOT) problems that involve conditional expectations in the objective functional. En route general results concerning optimization over (martingale) measures are proved that appear much more widely applicable. In particular the analysis leads us to introduce a notion of so-called curtain transports; in a main contribution we highlight the rôle that these transports play in (non-linear) MOT.

math.PR↗