Sergio Angelo Albeverio

• Prof.Dr.Dr.h.c.
• University of Bonn

The general framework on the non-local Markovian symmetric forms on weighted $l^p$ $(p \in [1, \infty])$ spaces constructed by [A,Kagawa,Yahagi,Y 2020], by restricting the situation where $p =2$, is applied to such measure spaces as the space cut-off $P(\phi)_2$ Euclidean quantum field, the $2$-dimensional Euclidean quantum fields with exponential...

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's op...

Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non-t...

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 coun...

General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {\cal B}(S), \mu)$, with $S$ Fr{\'e}chet spaces such that $S \subset {\mathbb R}^{\mathbb N}$, ${\cal B}(S)$ is the Borel $\sigma$-field of $S$, and $\mu$ is a Borel probability measure on $S$, are introduced. Firstly, a family...

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the $x$-direction and, moreover, in the presence of an impurity modelled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian imp...

We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified...

We introduce a stochastic analysis of Grassmann random variables suitable for the stochastic quantization of Euclidean fermionic quantum field theories. Analysis on Grassmann algebras is developed from the point of view of quantum probability: a Grassmann random variable is a homomorphism of an abstract Grassmann algebra into a quantum probability...

We study a family of McKean-Vlasov type ergodic optimal control problems with linear control, and quadratic dependence on control of the cost function. For this class of problems we establish existence and uniqueness of optimal control. We propose an $N$-particles Markovian optimal control problem approximating the McKean-Vlasov one and we prove th...

A stochastic approach to the (generic) mean-field limit in Bose–Einstein Condensation is described and the convergence of the ground-state energy as well as of its components are established. For the one-particle process on the path space, a total variation convergence result is proved. A strong form of Kac’s chaos on path-space for the [Formula: s...

We consider a stochastic Camassa-Holm equation driven by a one-dimensional Wiener process with a first order differential operator as diffusion coefficient. We prove the existence and uniqueness of local strong solutions of this equation. In order to do so, we transform it into a random quasi-linear partial differential equation and apply Kato's op...

We consider stochastic differential equations with a drift term of gradient type and driven by Gaussian white noise on [Formula: see text]. Particular attention is given to the kernel [Formula: see text], [Formula: see text] of the transition semigroup associated with the solution process. Under some rather strong regularity and growth assumptions...

In this note we consider a quantum mechanical particle moving inside an infinitesimally thin layer constrained by a parabolic well in the x-direction and, moreover, in the presence of an impurity modeled by an attractive Gaussian potential. We investigate the Birman-Schwinger operator associated to a model assuming the presence of a Gaussian impuri...

A Feynman path integral formula for the Schr\"odinger 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 co...

We study a class of elliptic SPDEs with additive Gaussian noise on $\mathbb{R}^2 \times M$, with $M$ a $d$-dimensional manifold equipped with a positive Radon measure, and a real-valued non linearity given by the derivative of a smooth potential $V$, convex at infinity and growing at most exponentially. For quite general coefficients and a suitable...

In this paper, we study the small noise asymptotic expansions for certain classes of local volatility models arising in finance. We provide explicit expressions for the involved coefficients as well as accurate estimates on the remainders. Moreover, we perform a detailed numerical analysis, with accuracy comparisons, of the obtained results by mean...

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general cadlag semimartingales taking values in Lie groups are defined and investigated. The considered set of SDEs, first introduced by S. Cohen, includes affine and Marcus type SDEs as well as smooth SDEs driven by Levy processes and iterated random maps....

A stochastic approach to the (generic) mean-field limit in Bose-Einstein Condensation is described and the convergence of the ground state energy as well as of its components are established. For the one-particle process on the path space a total variation convergence result is proved. A strong form of Kac's chaos on path-space for the $k$-particle...

Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non t...

We prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\mathbb{R}^2$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension with a Gibbs measure in $d$ dimensions. This p...

We consider stochastic differential equations driven by Gaussian white noise on $\R^d$. % We provide applications to models for financial %markets.\\ Particular attention is given to the kernel $p_t,\,t\geq 0$ of the transition semigroup associated with the solution process.\\ Under some assumptions on the coefficients, we prove that the small time...

In this paper we study the small noise asymptotic expansions for certain classes of local volatility models arising in finance. We provide explicit expressions for the involved coefficients as well as accurate estimates on the remainders. Moreover, we perform a detailed numerical analysis, with accuracy comparisons, of the obtained results by mean...

Techniques of infinite dimensional integrations providing the mathematical definition of Feynman path integrals are shortly reviewed and generalized, and then applied to the construction of a functional integral representation for solutions of a general class of higher-order heat-type equations.

We consider a bounded block operator matrix of the form $$ L=\left(\begin{array}{cc} A & B \\ C & D \end{array} \right), $$ where the main-diagonal entries $A$ and $D$ are self-adjoint operators on Hilbert spaces $H_{_A}$ and $H_{_D}$, respectively; the coupling $B$ maps $H_{_D}$ to $H_{_A}$ and $C$ is an operator from $H_{_A}$ to $H_{_D}$. It is a...

We give a direct construction of invariant measures and global flows for the stochastic quantization equation to the quantum field theoretical $\Phi ^4_3$-model on the $3$-dimensional torus. This stochastic equation belongs to a class of singular stochastic partial differential equations (SPDEs) presently intensively studied, especially after Haire...

Stochastic symmetries and related invariance properties of finite dimensional SDEs driven by general c\`adl\`ag semimartingales taking values in Lie groups are defined and investigated. In order to enlarge the class of possible symmetries of SDEs, the new concepts of gauge and time symmetries for semimartingales on Lie groups are introduced. Markov...

We prove the entropy-chaos property for the system of N undistinguishable interacting diffusions rigorously associated with the ground state of N trapped Bose particles in the Gross-Pitaevskii scaling limit of infinite particles. On the path-space we show that the sequence of probability measures of the one-particle interacting diffusion weakly con...

We study the scattering problem for Schrödinger operators in impedance form, with step-like impedance functions. In this paper, the case where the jumps are located at the vertices of periodic lattice is considered. The set of reflection coefficients is completely described and the scattering map is shown to be homeomorphic. Also, an algorithm reco...

In this note, we continue our analysis of the behavior of selfadjoint Hamiltonians with symmetric double wells given by twin point interactions perturbing various types of “free Hamiltonians” as the distance between the two centers shrinks to zero. In particular, by making the coupling constant to be renormalized and also dependent on the separatio...

The existence of a weak solution to a McKean–Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under suitable hypotheses. The distribution of any solution to the system at each fixed time is shown to be unique, when the density exists. The existence of a probability density...

We study a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by Lévy noise. We define a Hilbert-Banach setting in which we prove existence and uniqueness of solutions under general assumptions on the drift and the Ĺevy noise. We then prove a decomposition of the solution process into a stationary...

Collecting together contributed lectures and mini-courses, this book details the research presented in a special semester titled “Geometric mechanics – variational and stochastic methods” run in the first half of 2015 at the Centre Interfacultaire Bernoulli (CIB) of the Ecole Polytechnique Fédérale de Lausanne. The aim of the semester was to develo...

We use Yosida approximation to find an It\^o formula for mild solutions $\left\{X^x(t), t\geq 0\right\}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a L\'evy process. The functions to which we apply such It\^o formula are in $C^{1,2}([0,T]\time...

We consider two unitary representations of the infinite-dimensional groups of smooth paths with values in a compact Lie group. The first representation is induced by quasi-invariance of the Wiener measure, and the second representation is the energy representation. We define these representations and their basic properties, and then we prove that t...

Spectral theory, Scattering, Quantum transport, Quantum communications and computations.

In this note we continue our analysis (started in [1]) of the isotropic three-dimensional harmonic oscillator perturbed by a pair of identical attractive point interactions symmetrically situated with respect to the origin, that is to say the mathematical model describing a symmetric quantum dot with a pair of point impurities. In particular, by ma...

We describe a class of explicit invariant measures for both finite and infinite dimensional Stochastic Differential Equations (SDE) driven by L\'evy noise. We first discuss in details the finite dimensional case with a linear, resp. non linear, drift. In particular, we exhibit a class of such SDEs for which the invariant measures are given in expli...

It is proven that the relativistic quantum fields obtained from analytic continuation of convoluted generalized (L\'evy type) noise fields have positive metric, if and only if the noise is Gaussian. This follows as an easy observation from a criterion by K. Baumann, based on the Dell'Antonio-Robinson-Greenberg theorem, for a relativistic quantum fi...

In this presentation we wish to provide an overview of the spectral features of the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-interaction centred at the origin or by a pair of identical attractive δ-interactions symmetrically situated with respect to the origin. Given t...

In the present paper we study the dependence of fractal and metric properties of numbers which are non-normal resp. essentially non-normal w.r.t. a chosen system of numeration. In particular, we solve open problems mentioned in [1] and prove that there exist expansions (the Q⁎-expansions or Q⁎-representations) for real numbers such that the corresp...

We consider a J-self-adjoint 2x2 block operator matrix L in the Feshbach spectral case, that is, in the case where the spectrum of one main-diagonal entry is embedded into the absolutely continuous spectrum of the other main-diagonal entry. We work with the analytic continuation of one of the Schur complements of L to the unphysical sheets of the s...

The interaction between quantum mechanics, quantum field theory, stochastic partial differential equations and infinite dimensional analysis is briefly surveyed, referring in particular to models and techniques to which L. Streit has given outstanding contributions.

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive d¢- interaction, of strength b, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a ‘coupling constant renormalisation’. Here we get the H...

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ'-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a ‘coupling constant renormalisation’. Here we get the Ha...

We revisit von Neumann’s determination of the representations of the canonical commutation relations in Weyl form. We present an exposition of the original insights set within the convenient notational framework of symplectic structures. We study von Neumann’s projection operator and show how the complex phase space representation arises.

We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $G\times G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In particular, we find a condition on $f$ under which the corresponding algebra is a Leibniz algebra. Moreover,...

The relative partition function and the relative zeta function of the perturbation of the Laplace operator by a Coulomb potential plus a point interaction centered in the origin is discussed. Applications to the study of the Casimir effect are indicated.

The theory of infinite dimensional oscillatory integrals and some of its applications are discussed, with special attention to the relations with the original work of K. Itô in this area. A recent general approach to infinite dimensional integration which unifies the case of oscillatory integrals and the case of probabilistic type integrals is pres...

In this presentation we wish to provide an overview of the spectral features of the self-adjoint Hamiltonian of the three-dimensional isotropic harmonic oscillator perturbed by either a single attractive δ-interaction centred at the origin or by a pair of identical attractive δ-interactions symmetrically situated with respect to the origin. Given t...

We study properties of the distribution of a random variable of the continued fraction form where are independent and not necessarily identically distributed random variables. We prove the singularity of and study the fine spectral structure of such measures.

We perform a Doob h-transformation of the N-body Hamiltonian for N trapped interacting Bose particles starting from the unique real strictly positive ground state. This identifies the unitarily equivalent Markov generator describing the N interacting diffusion processes. By applying the convergence results coming from the Gross–Pitaevskii (GP) scal...

This chapter discusses the general mathematical properties of PT-symmetric operators within the Krein spaces framework, focusing on the aspects of the Krein spaces theory that may be more appealing to mathematical physicists. Every physically meaningful PT-symmetric operator should be a self adjoint operator in a suitably chosen Krein space and a p...

We establish several new fractal and number theoretical phenomena connected with expansions which are generated by infinite linear iterated function systems. First of all we show that the systems $\Phi$ of cylinders of generalized L\"uroth expansions are, generally speaking, not faithful for the Hausdorff dimension calculation. Using Yuval Peres' a...

In this note we provide an alternative way of defining the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ'-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), that was rigorously defined in a previous paper by means of a " coupling constant renormalisation ". Here we get the...

We establish several new probabilistic, dynamical, dimensional and number theoretical phenomena connected with Ostrogradsky-Sierpi\'nski-Pierce expansion. First of all, we develop metric, ergodic and dimensional theories of the Ostrogradsky-Sierpi\'nski-Pierce expansion. In particular, it is proven that for Lebesgue almost all real numbers any digi...

We consider the second Ostrogradsky expansion from the number theory, probability theory, dynamical systems and fractal geometry points of view, and establish several new phenomena connected with this expansion. First of all we prove the singularity of the random second Ostrogradsky expansion. Secondly we study properties of the symbolic dynamical...

We rigorously define the self-adjoint one-dimensional Salpeter Hamiltonian perturbed by an attractive interaction, of strength centred at the origin, by explicitly providing its resolvent. Our approach is based on a “coupling constant renormalisation”, a technique used first heuristically in quantum field theory and implemented in the rigorous math...

We discuss the probabilistic representation of the solutions of the heat equation perturbed by a repulsive point interaction in terms of a perturbation of Brownian motion, via a Feynman-Kac formula involving a local time functional. An application to option pricing is given, interpolating between the extreme cases of classical Black-Scholes options...

In many areas of mathematics both oscillatory and probabilistic type infinite-dimensional integrals arise. It is well known for the corresponding finite-dimensional theory that these integrals have common aspects but also strong differences. We present an introduction to certain common aspects of these integration theories and mention some applicat...

“Knock-out options relate to ordinary options the way crack relates to cocaine”, George Soros. Fig. 2 Graphical comparison between the new put introduced in our paper (lambda=2) and the corresponding knock-out put ("up and out", lambda=+∞), both having the exercise price at K=1, which is also the location of the "permeable barrier" (with sigma^2=r=...

“Knock-out options relate to ordinary options the way crack relates to cocaine”, George Soros. Fig. 3 Magnification in the neighbourhood of the strike price of the graphical comparison between the new put introduced in our paper (lambda=2) and the corresponding knock-out put ("up and out", lambda=+∞), both having the exercise price at K=1, which is...

“Knock-out options relate to ordinary options the way crack relates to cocaine”, George Soros. Fig. 2 Graphical comparison between the new put introduced in our paper (lambda=2) and the corresponding knock-out put ("up and out", lambda=+∞), both having the exercise price at K=1, which is also the location of the "permeable barrier" (with sigma^2=r...

“Knock-out options relate to ordinary options the way crack relates to cocaine”, George Soros. Fig. 3 Magnification in the neighbourhood of the strike price of the graphical comparison between the new put introduced in our paper (lambda=2) and the corresponding knock-out put ("up and out", lambda=+∞), both having the exercise price at K=1, which i...

“Knock-out options relate to ordinary options the way crack relates to cocaine”, George Soros. Fig. 1 Graphical comparison between the new put introduced in our paper (lambda=2) and the classical Black-Scholes vanilla put (lambda=0), both having the exercise price at K=1, which is also the location of the "permeable barrier" (with sigma^2=r=0.04,...

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...

We overview the recent results on the shift of the spectrum and norm bounds for variation of spectral subspaces of a Hermitian operator under an additive Hermitian perturbation. Along with the known results, we present a new subspace variation bound for the generic off-diagonal subspace perturbation problem. We also demonstrate how some of the abst...

We discuss the probabilistic representation of the solutions of the heat equation perturbed by a repulsive point interaction in terms of a perturbation of Brownian motion, via a Feynman-Kac formula involving a local time functional. An application to option pricing is given, interpolating between the extreme cases of classical Black-Scholes options...

After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple exa...

Through the consideration of a homeomorphism between C([0, T] → ℝℤ) and C([0, T] → (S1)ℤ), a one-to-one correspondence between stochastic prosesses with values in ℝℤ resp. (S1)ℤ is discussed. In particular, uniqueness of solutions of SDE's and contraction properties of the semigroups of the corresponding Markov processes on these spaces are studied...

Real ideals of compact operators for (complex) factors are investigated. A description (up to isomorphisms) of real two-sided ideals of relatively compact operators of a complex W*-factors is given. A relative weak (RW) $_r$ convergence in a real Hilbert space is introduced. The classical Hilbert characterization of compactness of operators is ext...

Wavelet coefficients of a process have arguments shift and scale. It can thus be viewed as a time series along shift for each scale. We have considered in the previous study general wavelet coefficient domain estimators and revealed a localization property with respect to shift. In this paper, we formulate the localization property with respect to...

Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The D...

Asymptotic expansions are derived as power series in a small coefficient entering a nonlinear multiplicative noise and a deterministic driving term in a nonlinear evolution equation. Detailed estimates on remainders are provided.

We stu\dd y a class of nonlinear stochastic partial differential equations with dissipative nonlinear drift, driven by L\'evy noise. Our work is divided in two parts. In the present part I we first define a Hilbert-Banach setting in which we can prove existence and uniqueness of solutions under general assumptions on the drift and the L\'evy noise....

We review recent progress in the direct and inverse scattering theory for one-dimensional Schrödinger operators in impedance form. Two classes of non-smooth impedance functions are considered. Absolutely continuous impedances correspond to singular Miura potentials that are distributions from W 2, loc -1 (ℝ); nevertheless, most of the classic scatt...

We introduce a rigorous mathematical model of abelian quantized Chern- Simons theory (C.S. theory) based on the theory of infinite-dimensional oscillatory integrals developed by the first author and Høegh-Krohn. We construct a gauge-fixed C.S. path integral as a Fresnel integral in a certain Hilbert space. Wilson loop variables are defined as Fresn...

We study general (not necessarily Hamiltonian) first-order symmetric system J y′(t)−B(t)y(t) = Δ(t) f(t) on an interval \({\mathcal{I}=[a,b) }\) with the regular endpoint a. It is assumed that the deficiency indices n ±(T min) of the minimal relation T min associated with this system in \({L^2_\Delta(\mathcal{I})}\) satisfy \({n_-(T_{\rm min})\leq...

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ'-interaction, of strength β, centred at 0 (the bottom of the confining parabolic potential), by explicitly providing its resolvent. Our approach is based on a 'coupling constant renormalization', related to a technique originated in quantum fie...

We study existence and uniqueness of an invariant measure for infinite dimensional stochastic differential equations with dissipative polynomially bounded nonlinear terms. We also exhibit the existence of a density with respect to a Gaussian measure. Moreover, we decompose the solution process into a stationary component and a component which vanis...

We present a brief biographical review of the scientific work and achievements of Vladimir M. Shelkovich on the occasion of his sudden death in February 2013.

We study a reaction–diffusion evolution equation perturbed by a space–time Lévy noise. The associated Kolmogorov operator is the sum of the infinitesimal generator of a C0C0-semigroup of strictly negative type acting on a Hilbert space and a nonlinear term which has at most polynomial growth, is non necessarily Lipschitz and is such that the whole...

We study families $\Phi$ of coverings which are faithful for the Hausdorff dimension calculation on a given set $E$ (i. e., special relatively narrow families of coverings leading to the classical Hausdorff dimension of an arbitrary subset of $E$) and which are natural generalizations of comparable net-coverings. They are shown to be very useful fo...

We investigate spectral properties of spherical Schrödinger operators (also known as Bessel operators) with δ-point interactions concentrated on a discrete set. We obtain necessary and sufficient conditions for these Hamiltonians to be self-adjoint, lower-semibounded and also we investigate their spectra. We also extend the classical Bargmann estim...

We rigorously define the self-adjoint Hamiltonian of the harmonic oscillator perturbed by an attractive δ '− interaction centred at 0 (the bottom of the confining parabolic potential), formally written as.... by explicitly providing its resolvent. Our approach is based on a “coupling constant renormalisation”, related to a technique originated in q...

We study the dynamics of an arbitrary (2,1)(2,1)-rational function f(x)=(x2+ax+b)/(cx+d)f(x)=(x2+ax+b)/(cx+d) on the field CpCp of complex pp-adic numbers. We show that the pp-adic dynamical system generated by ff has a very rich behavior. Siegel disks may either coincide or be disjoint for different fixed points of the dynamical system. Also, we f...

We study the long-time asymptotics of continuous-time branching random walk on ℤd (d ≥ 1) with a single source (i.e., branching site). The random walk is assumed homogeneous, symmetric, irreducible, and having zero mean and finite variance of jumps. We find the limiting extinction probability and the asymptotics of all integer moments for the total...