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Universal linear manipulation via routing and projective measurements
Authors:
Alessio Baldazzi,
Sonia Mazzucchi,
Lorenzo Pavesi
Abstract:
Multiport interferometers with $N$ ports are basic devices in both classical and quantum photonics. Ideally, they implement a linear unitary transformation between the input and output electric field vectors with $N$ components, each associated with a spatial mode of classical coherent light or a single photon. Standard designs for a fully reconfigurable universal multiport interferometer are give…
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Multiport interferometers with $N$ ports are basic devices in both classical and quantum photonics. Ideally, they implement a linear unitary transformation between the input and output electric field vectors with $N$ components, each associated with a spatial mode of classical coherent light or a single photon. Standard designs for a fully reconfigurable universal multiport interferometer are given by the Reck or the Clements schemes. In this work, we introduce routing schemes to implement a generic unitary transformation on classical coherent light or single photons using linear or tree geometries via multiple projective measurements on a single detector with the minimum number of components. Then, we generalize this result to the case of any multi-photon state for scattershot boson sampling experiments with multi-routing schemes. Finally, we test the robustness of routing schemes compared to universal schemes with respect to losses and phase noise.
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Submitted 5 August, 2026;
originally announced August 2026.
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A resource-efficient and noise-robust entanglement witness based on the swap test
Authors:
Sebastiano Guaraldo,
Sonia Mazzucchi,
Alessio Baldazzi,
Stefano Azzini,
Lorenzo Pavesi
Abstract:
Quantum entanglement is an essential resource for quantum technologies, and the controlled swap test provides a versatile tool for its detection and quantification. Here, we propose a SWAP-based entanglement witness that applies to arbitrary two-qubit states - both pure and mixed - and provides a lower bound on the concurrence. The method is resource-efficient, robust to noise, and platform-indepe…
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Quantum entanglement is an essential resource for quantum technologies, and the controlled swap test provides a versatile tool for its detection and quantification. Here, we propose a SWAP-based entanglement witness that applies to arbitrary two-qubit states - both pure and mixed - and provides a lower bound on the concurrence. The method is resource-efficient, robust to noise, and platform-independent. As an example, we validate the approach on a room-temperature photonic chip, where the swap test is carried out using only linear and well-established integrated optical components. The robustness of the method against photonic-hardware noise is also analysed. Our results establish a simple and reliable tool for entanglement witnessing.
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Submitted 23 December, 2025;
originally announced December 2025.
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Chernoff solutions of the heat and the Schrödinger equation in the Heisenberg group
Authors:
Nicolò Drago,
Sonia Mazzucchi,
Andrea Pinamonti
Abstract:
This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schrödinger equations on the Heisenberg group $\mathbb{H}^d$. Using semigroup approximation techniques, we obtain analytically tractable and numerically implementable representations of fundamental solutions. In particular, we establish a new connection between the heat equa…
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This paper investigates the application of the classical Chernoff's theorem to construct explicit solutions for the heat and Schrödinger equations on the Heisenberg group $\mathbb{H}^d$. Using semigroup approximation techniques, we obtain analytically tractable and numerically implementable representations of fundamental solutions. In particular, we establish a new connection between the heat equation and Brownian motion on $\mathbb{H}^d$ and provide a rigorous realization of the Feynman path integral for the Schrödinger equation. The study highlights the challenges posed by the noncommutative structure of the Heisenberg group and opens new directions for PDEs on sub-Riemannian manifolds.
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Submitted 24 March, 2025;
originally announced March 2025.
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Phase space analysis of higher-order dispersive equations with point interactions
Authors:
Sonia Mazzucchi,
Fabio Nicola,
S. Ivan Trapasso
Abstract:
We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory fun…
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We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave packets) of the corresponding fundamental solution - in fact, locating the modulation/amalgam space regularity of such generalized Fresnel-type oscillatory functions is a problem of independent interest in harmonic analysis.
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Submitted 22 July, 2024;
originally announced July 2024.
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Phase space analysis of finite and infinite dimensional Fresnel integrals
Authors:
Sonia Mazzucchi,
Fabio Nicola,
S. Ivan Trapasso
Abstract:
The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class $M^{\infty,1}$ - a family of continuous and bounded functions, locall…
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The full characterization of the class of Fresnel integrable functions is an open problem in functional analysis, with significant applications to mathematical physics (Feynman path integrals) and the analysis of the Schrödinger equation. In finite dimension, we prove the Fresnel integrability of functions in the Sjöstrand class $M^{\infty,1}$ - a family of continuous and bounded functions, locally enjoying the mild regularity of the Fourier transform of an integrable function. This result broadly extends the current knowledge on the Fresnel integrability of Fourier transforms of finite complex measures, and relies upon ideas and techniques of Gabor wave packet analysis. We also discuss the problem of designing infinite-dimensional extensions of this result, obtaining the first, non-trivial concrete realization of a general framework of projective functional extensions introduced by Albeverio and Mazzucchi. As an interesting byproduct, we obtain the exact $M^{\infty,1} \to L^\infty$ operator norm of the free Schrödinger evolution operator.
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Submitted 7 February, 2025; v1 submitted 29 March, 2024;
originally announced March 2024.
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Histogram-less LiDAR through SPAD response linearization
Authors:
Alessandro Tontini,
Sonia Mazzucchi,
Roberto Passerone,
Nicolò Broseghini,
Leonardo Gasparini
Abstract:
We present a new method to acquire the 3D information from a SPAD-based direct-Time-of-Flight (d-ToF) imaging system which does not require the construction of a histogram of timestamps and can withstand high flux operation regime. The proposed acquisition scheme emulates the behavior of a SPAD detector with no distortion due to dead time, and extracts the Tof information by a simple average opera…
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We present a new method to acquire the 3D information from a SPAD-based direct-Time-of-Flight (d-ToF) imaging system which does not require the construction of a histogram of timestamps and can withstand high flux operation regime. The proposed acquisition scheme emulates the behavior of a SPAD detector with no distortion due to dead time, and extracts the Tof information by a simple average operation on the photon timestamps ensuring ease of integration in a dedicated sensor and scalability to large arrays. The method is validated through a comprehensive mathematical analysis, whose predictions are in agreement with a numerical Monte Carlo model of the problem. Finally, we show the validity of the predictions in a real d-ToF measurement setup under challenging background conditions well beyond the typical pile-up limit of 5% detection rate up to a distance of 3.8 m.
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Submitted 13 October, 2023;
originally announced October 2023.
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Feynman path integrals on compact Lie groups with bi-invariant Riemannian metrics
Authors:
Nicoló Drago,
Sonia Mazzucchi,
Valter Moretti
Abstract:
In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral, and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant…
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In this work we consider a suitable generalization of the Feynman path integral on a specific class of Riemannian manifolds consisting of compact Lie groups with bi-invariant Riemannian metrics. The main tools we use are the Cartan development map, the notion of oscillatory integral, and the Chernoff approximation theorem. We prove that, for a class of functions of a dense subspace of the relevant Hilbert space, the Feynman map produces the solution of the Schrödinger equation, where the Laplace-Beltrami operator coincides with the second order Casimir operator of the group.
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Submitted 28 August, 2025; v1 submitted 6 July, 2023;
originally announced July 2023.
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Generation of quantum-certified random numbers using on-chip path-entangled single photons from an LED
Authors:
Nicolò Leone,
Stefano Azzini,
Sonia Mazzucchi,
Valter Moretti,
Matteo Sanna,
Massimo Borghi,
Gioele Piccoli,
Martino Bernard,
Mher Ghulinyan,
Lorenzo Pavesi
Abstract:
Single-photon entanglement is a peculiar type of entanglement in which two or more degrees of freedom of a single photon are correlated quantum-mechanically. Here, we demonstrate a photonic integrated chip (PIC) able to generate and manipulate single-photon path-entangled states, using a commercial red LED as light source. A Bell test, in the Clauser, Horne, Shimony and Holt (CHSH) form, is perfor…
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Single-photon entanglement is a peculiar type of entanglement in which two or more degrees of freedom of a single photon are correlated quantum-mechanically. Here, we demonstrate a photonic integrated chip (PIC) able to generate and manipulate single-photon path-entangled states, using a commercial red LED as light source. A Bell test, in the Clauser, Horne, Shimony and Holt (CHSH) form, is performed to confirm the presence of entanglement, resulting in a maximum value of the CHSH correlation parameter equal to $2.605 \pm 0.004$. This allows us to use it as an integrated semi-device independent quantum random number generator able to produce certified random numbers. The certification scheme is based on a Bell's inequality violation and on a partial characterization of the experimental setup, without the need of introducing any further assumptions either on the input state or on the particular form of the measurement observables. In the end a min-entropy of $33\%$ is demonstrated.
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Submitted 27 March, 2023;
originally announced March 2023.
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Entropy certification of a realistic QRNG based on single-particle entanglement
Authors:
Sonia Mazzucchi,
Nicolò Leone,
Stefano Azzini,
Lorenzo Pavesi,
Valter Moretti
Abstract:
In single-particle entanglement (SPE) two degrees of freedom of a single particle are entangled. SPE is a resource that can be exploited both in quantum communication protocols and in experimental tests of noncontextuality based on the Kochen-Specker theorem. SPE can be certified via a test of quantum contextuality based on Bell inequalities. Experiments of Bell-like inequality violation by single…
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In single-particle entanglement (SPE) two degrees of freedom of a single particle are entangled. SPE is a resource that can be exploited both in quantum communication protocols and in experimental tests of noncontextuality based on the Kochen-Specker theorem. SPE can be certified via a test of quantum contextuality based on Bell inequalities. Experiments of Bell-like inequality violation by single particle entangled systems may be affected by an analogue of the locality loophole in this context, due to the presence of unavoidable non-idealities in the experimental devices which actually produce unwanted correlations between the two observables that are simultaneously measured. This issue is tackled here by quantitatively analyzing the behaviour of realistic devices in SPE experiments with photons. In particular, we show how it is possible to provide a semi-device independent randomness certification of realistic quantum random number generators based on Bell inequality violation by SPE states of photons. The analysis is further enlarged to encompass, with a Markovian model, memory effects due to dead time, dark counts and afterpulsing affecting single photon detectors, in particular when not dealing with coincidence measurements. An unbiased estimator is also proposed for quantum transition probabilities out of the collection of experimental data.
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Submitted 2 August, 2021; v1 submitted 13 April, 2021;
originally announced April 2021.
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Certified quantum random number generator based on single-photon entanglement
Authors:
Nicolò Leone,
Stefano Azzini,
Sonia Mazzucchi,
Valter Moretti,
Lorenzo Pavesi
Abstract:
Quantum entanglement represents an ideal resource to guarantee the security of random numbers employed in many scientific and cryptographic applications. However, entanglement-based certified random number generators are particularly challenging to implement. Here, we demonstrate a new certified quantum random number generator based on momentum-polarization entangled single photon states. The use…
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Quantum entanglement represents an ideal resource to guarantee the security of random numbers employed in many scientific and cryptographic applications. However, entanglement-based certified random number generators are particularly challenging to implement. Here, we demonstrate a new certified quantum random number generator based on momentum-polarization entangled single photon states. The use of single photon entanglement allows employing an attenuated laser source and a simple setup where only linear optical components are utilized. For the latter, a semi-device-independent modeling of the photonic quantum random number generator is developed, which certifies a minimum entropy of $(2.5\pm 0.5)\%$, corresponding to a generation rate of 4.4 kHz. At the expenses of a higher level of trust in the system, the certified minimum entropy can be increased to $(30.1 \pm0.5 )\%$, implying a generation rate of 52.7 kHz. Our results show that a simple optical implementation combined with an accurate modeling provide an entanglement-based high-security quantum random number generator using imperfect devices.
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Submitted 17 September, 2021; v1 submitted 9 April, 2021;
originally announced April 2021.
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Bell inequality violation by entangled single photon states generated from a laser, a LED or a Halogen lamp
Authors:
M. Pasini,
N. Leone,
S. Mazzucchi,
V. Moretti,
D. Pastorello,
L. Pavesi
Abstract:
In single-particle or intraparticle entanglement, two degrees of freedom of a single particle, e.g., momentum and polarization of a single photon, are entangled. Single-particle entanglement (SPE) provides a source of non classical correlations which can be exploited both in quantum communication protocols and in experimental tests of noncontextuality based on the Kochen-Specker theorem. Furthermo…
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In single-particle or intraparticle entanglement, two degrees of freedom of a single particle, e.g., momentum and polarization of a single photon, are entangled. Single-particle entanglement (SPE) provides a source of non classical correlations which can be exploited both in quantum communication protocols and in experimental tests of noncontextuality based on the Kochen-Specker theorem. Furthermore, SPE is robust under decoherence phenomena. Here, we show that single-particle entangled states of single photons can be produced from attenuated sources of light, even classical ones. To experimentally certify the entanglement, we perform a Bell test, observing a violation of the Clauser, Horne, Shimony and Holt (CHSH) inequality. On the one hand, we show that this entanglement can be achieved even in a classical light beam, provided that first-order coherence is maintained between the degrees of freedom involved in the entanglement. On the other hand, we prove that filtered and attenuated light sources provide a flux of independent SPE photons that, from a statistical point of view, are indistinguishable from those generated by a single photon source. This has important consequences, since it demonstrates that cheap, compact, and low power entangled photon sources can be used for a range of quantum technology applications.
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Submitted 6 November, 2020; v1 submitted 22 March, 2020;
originally announced March 2020.
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Chernoff approximations of Feller semigroups in Riemannian manifolds
Authors:
Sonia Mazzucchi,
Valter Moretti,
Ivan Remizov,
Oleg Smolyanov
Abstract:
Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compa…
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Chernoff approximations of Feller semigroups and the associated diffusion processes in Riemannian manifolds are studied. The manifolds are assumed to be of bounded geometry, thus including all compact manifolds and also a wide range of non-compact manifolds. Sufficient conditions are established for a class of second order elliptic operators to generate a Feller semigroup on a (generally non-compact) manifold of bounded geometry. A construction of Chernoff approximations is presented for these Feller semigroups in terms of shift operators. This provides approximations of solutions to initial value problems for parabolic equations with variable coefficients on the manifold. It also yields weak convergence of a sequence of random walks on the manifolds to the diffusion processes associated with the elliptic generator. For parallelizable manifolds this result is applied in particular to the representation of Brownian motion on the manifolds as limits of the corresponding random walks.
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Submitted 18 October, 2021; v1 submitted 16 February, 2020;
originally announced February 2020.
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An operational construction of the sum of two non-commuting observables in quantum theory and related constructions
Authors:
Nicolò Drago,
Sonia Mazzucchi,
Valter Moretti
Abstract:
The existence of a real linear-space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables $A,B$ is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of the…
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The existence of a real linear-space structure on the set of observables of a quantum system -- i.e., the requirement that the linear combination of two generally non-commuting observables $A,B$ is an observable as well -- is a fundamental postulate of the quantum theory yet before introducing any structure of algebra. However, it is by no means clear how to choose the measuring instrument of the composed observable $aA+bB$ ($a,b\in \mathbb{R}$) if such measuring instruments are given for the addends observables $A$ and $B$ when they are incompatible observables. A mathematical version of this dilemma is how to construct the spectral measure of $f(aA+bB)$ out of the spectral measures of $A$ and $B$. We present such a construction with a formula which is valid for generally unbounded selfadjoint operators $A$ and $B$, whose spectral measures may not commute, and a wide class of functions $f: \mathbb{R} \to \mathbb{C}$. We prove that, in the bounded case the Jordan product of $A$ and $B$ can be constructed with the same procedure out of the spectral measures of $A$ and $B$. The formula turns out to have an interesting operational interpretation and, in particular cases, a nice interplay with the theory of Feynman path integration and the Feynman-Kac formula.
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Submitted 23 September, 2020; v1 submitted 24 September, 2019;
originally announced September 2019.
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A rigorous mathematical construction of Feynman path integrals for the Schrödinger equation with magnetic field
Authors:
Sergio Albeverio,
Nicolò Cangiotti,
Sonia Mazzucchi
Abstract:
A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functi…
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A Feynman path integral formula for the Schrödinger equation with magnetic field is rigorously mathematically realized in terms of infinite dimensional oscillatory integrals. We show (by the example of a linear vector potential) that the requirement of the independence of the integral on the approximation procedure forces the introduction of a counterterm to be added to the classical action functional. This provides a natural explanation for the appearance of a Stratonovich integral in the path integral formula for both the Schrödinger and heat equation with magnetic field.
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Submitted 27 July, 2019;
originally announced July 2019.
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Notes on the Ogawa integrability and a condition for convergence in the multidimensional case
Authors:
Nicolò Cangiotti,
Sonia Mazzucchi
Abstract:
The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated, proving that it cannot hold in general without the introduction of a "renormalization term". Explicit examples are provided.
The Ogawa stochastic integral is shortly reviewed and formulated in the framework of abstract Wiener spaces. The condition of universal Ogawa integrability in the multidimensional case is investigated, proving that it cannot hold in general without the introduction of a "renormalization term". Explicit examples are provided.
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Submitted 5 September, 2018;
originally announced September 2018.
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Probabilistic representation formula for the solution of fractional high order heat-type equations
Authors:
Stefano Bonaccorsi,
Mirko D'Ovidio,
Sonia Mazzucchi
Abstract:
We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with…
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We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order $α\in (0,1)$.
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Submitted 10 October, 2017; v1 submitted 10 November, 2016;
originally announced November 2016.
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An Itô calculus for a class of limit processes arising from random walks on the complex plane
Authors:
Stefano Bonaccorsi,
Craig Calcaterra,
Sonia Mazzucchi
Abstract:
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.
Within the framework of the previous paper [8]. we develop a generalized stochastic calculus for processes associated to higher order diffusion operators. Applications to the study of a Cauchy problem, a Feynman-Kac formula and a representation formula for higher derivatives of analytic functions are also given.
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Submitted 17 March, 2016;
originally announced March 2016.
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Peano on definition of surface area
Authors:
Gabriele H. Greco,
Sonia Mazzucchi,
Enrico M. Pagani
Abstract:
In this paper we investigate the evolution of the concept of area in Peano's works, taking into account the main role played by Grassmann's geometric-vector calculus and Peano's theory on derivative of measures. Geometric (1887) and bi-vectorial (1888) Peano's approaches to surface area mark the development of this topic during the first half of the last century. In the sequel we will present some…
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In this paper we investigate the evolution of the concept of area in Peano's works, taking into account the main role played by Grassmann's geometric-vector calculus and Peano's theory on derivative of measures. Geometric (1887) and bi-vectorial (1888) Peano's approaches to surface area mark the development of this topic during the first half of the last century. In the sequel we will present some significative contributions on surface area that are inspired and/or closely related to Peano's definition.
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Submitted 8 December, 2014;
originally announced December 2014.
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A unified approach to infinite dimensional integration
Authors:
Sergio Albeverio,
Sonia Mazzucchi
Abstract:
An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Schrödi…
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An approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is presented. It provides a truly infinite dimensional construction of integrals as linear functionals, as much as possible independent of the underlying topological and measure theoretical structure. Various applications are given, including, next to Schrödinger and diffusion equations, also higher order hyperbolic and parabolic equations.
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Submitted 11 November, 2014;
originally announced November 2014.
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Infinite dimensional oscillatory integrals with polynomial phase and applications to high order heat-type equations
Authors:
Sonia Mazzucchi
Abstract:
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution of a general class of high order heat-type equations.
The definition of infinite dimensional Fresnel integrals is generalized to the case of polynomial phase functions of any degree and applied to the construction of a functional integral representation of the solution of a general class of high order heat-type equations.
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Submitted 16 June, 2016; v1 submitted 18 May, 2014;
originally announced May 2014.
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High Order Heat-type Equations and Random Walks on the Complex Plane
Authors:
Stefano Bonaccorsi,
Sonia Mazzucchi
Abstract:
A probabilistic construction for the solution of a general class of high order heat-type equations is constructed in terms of the scaling limit of random walks in the complex plane.
A probabilistic construction for the solution of a general class of high order heat-type equations is constructed in terms of the scaling limit of random walks in the complex plane.
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Submitted 25 February, 2014;
originally announced February 2014.
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Peano on derivative of measures, strict derivative of distributive set functions
Authors:
Gabriele H. Greco,
Sonia Mazzucchi,
Enrico M. Pagani
Abstract:
By retracing research on coexistent magnitudes (grandeurs coexistantes) by Cauchy (1841), Peano in "Applicazioni geometriche del calcolo infinitesimale" (1887) defines the "density" (strict derivative) of a "mass" (a distributive set function) with respect to a "volume" (a positive distributive set function), proves its continuity (whenever the strict derivative exists) and shows the validity of…
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By retracing research on coexistent magnitudes (grandeurs coexistantes) by Cauchy (1841), Peano in "Applicazioni geometriche del calcolo infinitesimale" (1887) defines the "density" (strict derivative) of a "mass" (a distributive set function) with respect to a "volume" (a positive distributive set function), proves its continuity (whenever the strict derivative exists) and shows the validity of the mass-density paradigm: "mass" is recovered from "density" by integration with respect to "volume". It is remarkable that Peano's strict derivative provides a consistent mathematical ground to the concept of "infinitesimal ratio" between two magnitudes, successfully used since Kepler. In this way the classical (i.e., pre-Lebesgue) measure theory reaches a complete and definitive form in Peano's Applicazioni geometriche.
A primary aim of the present paper is a detailed exposition of Peano's work of 1887 leading to the concept of strict derivative of distributive set functions and their use. Moreover, we compare Peano's work and Lebesgue's "La mesure des grandeurs" (1935): in this memoir Lebesgue, motivated by coexistent magnitudes of Cauchy, introduces a uniform-derivative of certain additive set functions, a concept that coincides with Peano's strict derivative. Intriguing questions are whether Lebesgue was aware of the contributions of Peano and which role is played by the notions of strict derivative or of uniform-derivative in today mathematical practice.
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Submitted 22 February, 2010;
originally announced February 2010.
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On the Observables Describing a Quantum Reference Frame
Authors:
S. Mazzucchi
Abstract:
A reference frame F is described by the element g of the Poincare' group P which connects F with a given fixed frame F_0. If F is a quantum frame, defined by a physical object following the laws of quantum physics, the parameters of g have to be considered as quantum observables. However, these observables are not compatible and some of them, namely the coordinates of the origin of F, cannot be…
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A reference frame F is described by the element g of the Poincare' group P which connects F with a given fixed frame F_0. If F is a quantum frame, defined by a physical object following the laws of quantum physics, the parameters of g have to be considered as quantum observables. However, these observables are not compatible and some of them, namely the coordinates of the origin of F, cannot be represented by self-adjoint operators. Both these difficulties can be overcome by considering a positive-operator-valued measure (POVM) on P, covariant with respect to the left translations of the group, namely a covariance system. We develop a construction procedure for this kind of mathematical structure. The formalism is also used to discuss the quantum observables measured with respect to a quantum reference frame.
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Submitted 13 June, 2000;
originally announced June 2000.