Searcharxiv⌕ Search

arXiv subjects

Jan van Neerven

Publications and source records attributed to Jan van Neerven.

At least 19 recordsLinked to original sources
arXiv2026

The bounds $\hbar_{p,X}\lesssim(\beta_{p,X})^2$ and $\beta_{p,X}\lesssim(\hbar_{p,X})^2$ are sharp

It was proved in the 1980s by Burkholder and Bourgain that, for any Banach space $X$ and $1<p<\infty$, the UMD$_p$ property for $X$ is equivalent to boundedness of the Hilbert transform on $L^p(\R;X)$, and that the UMD constant $\beta_{p,X}$ and the Hilbert transform constant $\hbar_{p,X}$ are related by the quadratic bounds \begin{equation*} \hbar_{p,X}\lesssim(\beta_{p,X})^2, \qquad \beta_{p,X}\lesssim(\hbar_{p,X})^2. \end{equation*} In this paper we present examples showing that both bounds are sharp. More precisely, we construct explicit $2^n$-dimensional Banach spaces for which the Hilbert transform constant grows like $n$ and the UMD constant like $\sqrt n$, and a second family with the reverse behaviour.

math.FA↗
arXiv2026

Dimension-free $H^\infty$-calculus of angle $<\pi/2$ for UMD-valued Ornstein--Uhlenbeck operators

Let $1<p<\infty$, let $X$ be a UMD Banach space, let $1<p<\infty$, and let $L_d$ be the generator of the Ornstein--Uhlenbeck semigroup $(P_d(t))_{t\geq 0}$ on $L^p(\mathbb R^d,\gamma_d;X)$. We prove that the operators $-L_d$ are $R$-sectorial with a common angle strictly smaller than $\pi/2$ and with bounds independent of $d$. Combining this with the Hieber--Pr\"uss transference theorem and the Kalton--Weis angle comparison, we deduce that the operators $-L_d$ admit bounded $H^\infty$-calculi of a common angle strictly smaller than $\pi/2$, again with dimension-free bounds. The corresponding Walsh $R$-analyticity estimates are proved first and transferred to the Ornstein--Uhlenbeck setting by a central limit argument.

math.FA↗
arXiv2026

Necessary conditions for deterministic and stochastic maximal regularity

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

math.FA↗
arXiv2026

A Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces

We prove a Bismut-Elworthy-Li formula for linear stochastic evolution equations in Banach spaces. The admissible directions are those for which the deterministic orbit is square integrable with values in the Hilbertian range of the noise coefficient. Under this square function condition, the derivative of the semigroup is represented by a stochastic integral against an explicit deterministic control. The result is a concrete control realisation of the Cameron--Martin formula for the fixed-time Gaussian transition measures.

math.PR↗
arXiv2025

Holevo classicalisation and context integration: Global versus protocolwise integration

Suppose that a family of commutative measurement contexts is given, together with the corresponding classical probability laws. We study the problem under what conditions these contextwise descriptions can be combined into a single classical representation. We distinguish two ways in which this can be done. In {\em global integration}, the prescribed observable algebras are required to fit into a single commutative context and the corresponding probability laws to arise as marginals of one joint law. In {\em protocolwise integration}, by contrast, the context label is retained as part of the description, and the contextwise laws appear as conditional distributions within a single model of the measurement protocol. We show that global integration involves two distinct notions of compatibility: operator-algebraic compatibility of the underlying observables and probabilistic compatibility of their classicalised laws. The former implies the latter, but the converse fails; under the topological hypotheses stated below, probabilistic compatibility is characterised by a Kellerer-type marginal criterion. Protocolwise integration, on the other hand, always exists and is represented operator-algebraically by a direct-sum construction. When global integration exists, the two constructions arise as reductions of a common commutative refinement. We illustrate the distinction in the Bell--CHSH experiment, where Fine's theorem relates global probabilistic integration to Bell locality and the CHSH inequalities. This shows that the obstruction revealed by Bell--CHSH concerns not classical representation as such, but classical representation preserving specified cross-context identifications.

quant-ph↗
arXiv2025

Indiscernibility of quantum states

This paper provides a systematic study of the operational idea that a quantum ``state'' is only defined up to what can be distinguished by a chosen family of observables. Concretely, any von Neumann algebra of observables $\mathscr{M}$ induces an equivalence relation on pure and mixed states by declaring two preparations indiscernible when they give identical statistics for every observable in $\mathscr{M}$. The corresponding quotient, the \emph{Holevo space} associated with $\mathscr{M}$, is the effective (relational) state space of the experiment, explicitly dependent on the observer's available measurements. We analyse the resulting geometry and topology of these quotients, and prove a context-complete classical representation theorem: for every von Neumann algebra $\mathscr{M}$ there is a canonical lift $a\mapsto \widehat a$ to bounded continuous functions on the Holevo space, reproducing expectation values pointwise. In the commutative case this reduces to ordinary probability theory on the joint spectrum. The framework is illustrated in explicit examples, including position measurements of a free particle and polarisation measurements in the qubit, EPR, and Bell settings. In particular, in the EPR scenario, Charlie's joint observable defines a simplex of joint outcome distributions, whereas the Alice/Bob marginal viewpoint collapses the effective description to a lower-dimensional space by ``forgetting'' the correlation parameter. We show that by varying the polariser settings, the indiscernibility classes become conjugated (and generically reshuffled), and different settings are typically incompatible at the level of observable algebras.

quant-ph↗
arXiv2024

L\'evy measures on Banach spaces

We establish an explicit characterisation of L\'evy measures on both $L^p$-spaces and UMD Banach spaces. In the case of $L^p$-spaces, L\'evy measures are characterised by an integrability condition, which directly generalises the known description of L\'evy measures on sequence spaces. The latter has been the only known description of L\'evy measures on infinite dimensional Banach spaces that are not Hilbert. L\'evy measures on UMD Banach spaces are characterised by the finiteness of the expectation of a random {\gamma}-radonifying norm. Although this description is more abstract, it reduces to simple integrability conditions in the case of $L^p$-spaces.

math.FA↗
arXiv2024

Thermal time as an unsharp observable

We show that the Connes-Rovelli thermal time associated with the quantum harmonic oscillator can be described as an (unsharp) observable, that is, as a positive operator valued measure. We furthermore present extensions of this result to the free massless relativistic particle in one dimension and to a hypothetical physical system whose equilibrium state is given by the noncommutative integral.

math-ph↗
arXiv2024

Delayed choice experiments: An analysis in forward time

In this article, we present a detailed analysis of two famous delayed choice experiments: Wheeler's classic gedanken-experiment and the delayed quantum eraser. Our analysis shows that the outcomes of both experiments can be fully explained on the basis of the information collected during the experiments using textbook quantum mechanics only. At no point in the argument, information from the future is needed to explain what happens next. In fact, more is true: for both experiments, we show, in a strictly mathematical way, that a modified version in which the time-ordering of the steps is changed to avoid the delayed choice leads to exactly the same final state. In this operational sense, the scenarios are completely equivalent in terms of conclusions that can be drawn from their outcomes.

quant-ph↗
arXiv2022

Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices

We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann-Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces $L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the Stone-von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.

math.FA↗
arXiv2021

Functional Analysis

This book is based on notes compiled over the many years I have been teaching the course "Applied Functional Analysis" in the first year of the Master programme at Delft University of Technology, for students with previous exposure to the essentials of Real Analysis and the theory of Lebesgue integration. It offers a comprehensive introduction to functional analysis, covering both the abstract theory and applications to spectral theory, boundary value problems, semigroup theory, and quantum mechanics. It starts with the basic results of the subject and progresses towards a treatment of several advanced topics not commonly found in functional analysis textbooks, including Fredholm theory, form methods, boundary value problems, semigroup theory, trace formulas and a mathematical treatment of states and observables in quantum mechanics.

math.FA↗
arXiv2021

Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

We prove a new Burkholder-Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $(S(t,s))_{0\leq s\leq T}$ is a $C_0$-evolution family of contractions on a $2$-smooth Banach space $X$ and $(W_t)_{t\in [0,T]}$ is a cylindrical Brownian motion on a probability space $(Ω,P)$, then for every $0<p<\infty$ there exists a constant $C_{p,X}$ such that for all progressively measurable processes $g: [0,T]\times Ω\to X$ the process $(\int_0^t S(t,s)g_sdW_s)_{t\in [0,T]}$ has a continuous modification and $$E\sup_{t\in [0,T]}\Big\| \int_0^t S(t,s)g_sdW_s \Big\|^p\leq C_{p,X}^p \mathbb{E} \Bigl(\int_0^T \| g_t\|^2_{γ(H,X)}dt\Bigr)^{p/2}.$$ Moreover, for $2\leq p<\infty$ one may take $C_{p,X} = 10 D \sqrt{p},$ where $D$ is the constant in the definition of $2$-smoothness for $X$. Our result improves and unifies several existing maximal estimates and is even new in case $X$ is a Hilbert space. Similar results are obtained if the driving martingale $g_tdW_t$ is replaced by more general $X$-valued martingales $dM_t$. Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$ du_t = A(t)u_tdt + g_tdW_t, \quad u_0 = 0,$$ Under spatial smoothness assumptions on the inhomogeneity $g$, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

math.PR↗
arXiv2021

Relational analysis of the Frauchiger--Renner paradox and interaction-free detection of records from the past

We present an analysis of the Frauchiger-Renner Gedankenexperiment from the point of view of the relational interpretation of quantum mechanics. Our analysis shows that the paradox obtained by Frauchiger and Renner disappears if one rejects promoting one agent's certainty to another agent's certainty when it cannot be validated by records from the past. A by-product of our analysis is an interaction-free detection scheme for the existence of such records.

quant-ph↗
arXiv2020

The Garrison-Wong quantum phase operator revisited

We revisit the quantum phase operator $Φ$ introduced by Garrison and Wong. Denoting by $N$ the number operator, we provide a detailed proof of the Heisenberg commutation relation $ΦN - NΦ= iI$ on the natural maximal domain $D(ΦN) \cap D(N Φ)$ as well as the failure of the Weyl commutation relations, and discuss some further interesting properties of this pair.

quant-ph↗
arXiv2019

Continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations

We prove convergence of the solutions X_n of semilinear stochastic evolution equations dX_n(t) = (A_nX(t) + F_n(t,X_n(t)))dt + G_n(t,X_n(t))dW_H(t), X_n(0) = x_n, on a Banach space B, driven by a cylindrical Brownian motion W_H in a Hilbert space H. We assume that the operators A_n converge to A and the locally Lipschitz functions F_n and G_n converge to the locally Lipschitz functions F and G in an appropriate sense. Moreover, we obtain estimates for the lifetime of the solution X of the limiting problem in terms of the lifetimes of the approximating solutions X_n. We apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions. The operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator.

math.PR↗
arXiv2018

Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time

In this paper, we introduce a Weyl functional calculus $a \mapsto a(Q,P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $ L = -Δ+ x\cdot \nabla$, and give a simple criterion for restricted $L^p$-$L^q$ boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of $L$. It allows us to recover, unify, and extend, old and new results concerning the boundedness of $\exp(-zL)$ as an operator from $L^p(\mathbb{R}^d,γ_α)$ to $L^q(\mathbb{R}^d,γ_β)$ for suitable values of $z\in \mathbb{C}$ with $\Re z>0$, $p,q\in [1,\infty)$, and $α,β>0$. Here, $γ_τ$ denotes the centred Gaussian measure on $\mathbb{R}^d$ with density $(2πτ)^{-d/2}\exp(-|x|^2/2τ)$.

math.FA↗