XI International Conference

Stochastic and Analytic Methods

in Mathematical Physics

Yerevan, Armenia

September 2–7, 2019

Organizers:

Institute of Mathematics of National Academy of Sciences of Armenia; AmericanUniversity of Armenia; Armenian Mathematical Union

Scientic Committee:

Carlo Boldrighini (Rome); Gianfausto Dell'Antonio (Rome/Trieste); PavelExner (Prague); Vadim A. Malyshev (Moscow); Bruno Nachtergaele (UC Davis);Boris Nahapetian (Yerevan); Suren Poghosyan (Yerevan); Yakov G. Sinai(Princeton/Moscow); Herbert Spohn (Munich); Anatoly M. Vershik(St.Petersburg); Valentin Zagrebnov (Marseille); Hans Zessin (Bielefeld); ElenaZhizhina (Moscow)

Local Organizing Committee:

Victor Arzumanian; Rafayel Barkhudaryan; Aram Hajian; Linda Khachatryan;Boris Nahapetian; Suren Poghosyan (Chair)

Sponsors:

International Association of Mathematical Physics; Springer; Annales HenriPoincare; University of Potsdam; Research Mathematics Fund (Armenia); ScienceCommittee MES RA

web-page: http://math.sci.am/conference/sammp2019

Robert Adol'fovich Minlos

Stochastic and Analytic Methods

in Mathematical Physics

This is the 11th event in the series of conferences devoted to the problemsof Modern Mathematical Physics. The conference will take place on September2-7, 2019 in Yerevan. The venue of the conference is the American University ofArmenia (40 Marshal Baghramian Ave., Yerevan).

The Institute of Mathematics of National Academy of Sciences of the Republicof Armenia has organized ten international conferences in Mathematical Physicswith the rst conference taking place in 1982 and the latest one in 2016 in Yerevan.

The present conference is dedicated to the memory of famous Soviet andRussian mathematician Robert Adol'fovich Minlos (28 February 1931 9 January2018).

Among the participants of previous conferences were many well-knownscientists, such as Sergio Albeverio, Roland L. Dobrushin, Gianfausto Dell'Antonio,Vadim A. Malyshev, Robert A. Minlos, Salvador Miracle-Sole, Alexander Polyakov,Yakov G. Sinai, Anatoly M. Vershik and others.

The previous conferences covered various topics of the Mathematical Physicsand have provided an excellent opportunity for exchange of ideas and informationand contributed to the overall development of modern Mathematical Physics inArmenia. The topics of the present conference include classical and quantumstatistical physics, quantum dynamics, mathematical methods in quantummechanics and applications of point processes in statistical physics.

We welcome the participants and wish them interesting discussions and pleasanttime in Armenia.

Local Organizing Committee

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PROGRAM

Monday, September 2

09:0009:45 Registration

09:4510:00 Opening

10:0010:40 Sylvie Roelly (Potsdam): Statistical mechanical approaches for

solving innite-dimensional SDEs with memory

10:4011:20 Coee break

11:2012:00 Daniel Ueltschi (Warwick): Extremal states decomposition in quan-

tum spin systems

12:0012:40 Elena Zhizhina (Moscow): Asymptotic pointwise estimates for heat

kernels of convolution type operators

12:4014:30 Lunch

14:3015:10 Pavel Exner (Prague): Topology makes the spectral picture richer:

quantum graph examples

15:1015:40 Coee break

15:40 16:20 Andrey Piatnitski (Moscow): Homogenization of non-symmetric

convolution type operators in periodic media

19:30 Welcome Party

Tuesday, September 3

09:0009:40 Gian Michele Graf (Zurich): Indirect measurements of a harmonic

oscillator

09:4010:20 ClaudeAlain Pillet (Toulon): Thermodynamics of repeated quan-

tum measurements

10:2010:50 Coee break

10:5011:30 Saidakhmat Lakayev (Samarkand): The threshold eects for the

twoparticle Hamiltonians on lattices

11:3012:10 Alexander Lykov (Moscow): Longtime behaviour of innite chain

of harmonic oscillators

12:1012:40 Linda Khachatryan (Yerevan): Direct and inverse problems in the

theory of description of random elds

12:4014:30 Lunch

14:30 Round Table with Coee break R.A.Minlos: Life, work, and legacy

(Ðîáåðò Àäîëüôîâè÷ Ìèíëîñ: æèçíü, òâîð÷åñòâî, ìàòåìàòè÷åñêîå íàñëå-

äèå) Chair: Valentin Zagrebnov (Marseille)

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Wednesday, September 4

09:00–09:40 Jesper Møller (Aalborg): Determinantal point processes andtheir usefulness in spatial statistics09:40–10:20 Sabine Jansen (Munich): Cluster expansions with renormalizedactivities and applications to colloids10:20–10:50 Coffee break10:50–11:30 Dimitrios Tsagkarogiannis (L’Aquila): Virial inversion and den-sity functionals11:30–12:10 David Dereudre (Lille): DLR equations and rigidity for theSine–beta process12:10–12:30 Pierre Houdebert (Potsdam): Sharp phase transition for theWidom–Rowlinson model12:30–14:30 Lunch14:30–15:00 Suren Poghosyan (Yerevan): A Characterization of the Gibbsprocess in terms of a given factorial measure15:00–15:20 Leonid Kolesnikov (Munich): Activity expansions for correlationfunctions: Characterizing the domain of absolute convergence15:20–16:00 Mathias Rafler (Berlin): Integration by parts for conditionedpoint processes16:00–16:30 Coffee break16:30 Round Table Classical continuous systems: Existence, uniquenessand phase transitions: What do we know today? Chair: Hans Zessin (Bielefeld)

Thursday, September 5

09:00 Sightseeing Tour starts in front of the Presidium of the NationalAcademy of Sciences of Armenia, 24 Marshal Baghramian Ave.19:30 Banquet

Friday, September 6

09:00–09:40 Carlo Boldrighini (Rome): Three–dimensional incompressibleNavier–Stokes Equations: complex blow–up and related real flows09:40–10:20 Eugene Pechersky (Moscow): Application of the large deviationstheory to a stochastic version of Hawking–Penrose black hole model: Large emis-sion regime10:20–10:50 Coffee break

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10:50–11:30 Armen Shirikyan (Cergy–Pontoise): Entropy production in vis-cous fluid flows11:30–11:50 Nikita Vvedenskaya (Moscow): New expressions for local largedeviations probability11:50–12:10 Ostap Hryniv (Durham): Phase separation and sharp large de-viations12:10–12:30 Alexander Zass (Potsdam): Existence of Gibbsian point mea-sures via entropy methods12:30–14:30 Lunch14:30–14:50 Rodolfo Figari (Napoli): Regularized Hamiltonians for a threeBoson system with zero–range interactions14:50–15:10 Carlo Lucheroni (Camerino): Machine learning econometrics15:10–15:30 Anatoly Aristov (Moscow): Exact solutions of a nonclassicalequation with a nonlinearity under the Laplace operator15:30–15:50 Kazimierz Rajchel (Cracow): Trygonometric approach to theSchrodinger equation as a general case of known solutions

Saturday, September 7

09:00–09:40 Volker Betz (Darmstadt): The phase transition for random loopmodels on trees09:40–10:00 Hayk Sukiasyan (Yerevan): Semi-recursive algorithm of piece-wise linear approximation of two–dimensional function by the method of worstsegment dividing10:00–10:20 Tigran Harutyunyan (Yerevan): On some inverse problems10:20–10:50 Coffee break10:50–11:10 Alla Kuznetsova (Kazan): Fell bundle over a local group11:10–11:30 Rytis Jursenas (Vilnius): The peak model for the triplet exten-sions and their transformations to the reference Hilbert space11:30–11:50 Daniel Parra (Tokyo): Low-energy asymptotic in perturbed pe-riodically twisted quantum waveguides11:50–12:10 Maryam Ramezani (Bojnord): Integral equations in mathemat-ical physics12:10–12:30 Closing

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ABSTRACTS

Anatoly Aristov (Moscow)Exact solutions of a nonclassical equation with a nonlinearity under theLaplace operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Volker Betz (Darmstadt)The phase transition for random loop models on trees . . . . . . . . . . . . . . . . . . . . . 11

Carlo Boldrighini (Rome)Three-dimensional incompressible Navier-Stokes Equations: complex blow-upand related real flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

David Dereudre (Lille)DLR equations and rigidity for the Sine-beta process . . . . . . . . . . . . . . . . . . . . . . 13

Pavel Exner (Prague)Topology makes the spectral picture richer: quantum graph examples . . . . . . . 14

Rodolfo Figari (Napoli)Regularized Hamiltonians for a three Boson system with zero-rangeinteractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Gian Michele Graf (Zurich)Indirect measurements of a harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Tigran Harutyunyan (Yerevan)On some inverse problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Pierre Houdebert (Potsdam)Sharp phase transition for the Widom–Rowlinson model . . . . . . . . . . . . . . . . . . . 18

Ostap Hryniv (Durham)Phase separation and sharp large deviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Sabine Jansen (Munich)Cluster expansions with renormalized activities and applications to colloids 20Rytis Jursenas (Vilnius)The peak model for the triplet extensions and their transformations to thereference Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

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Linda Khachatryan (Yerevan)Direct and inverse problems in the theory of description of random fields . . 22

Leonid Kolesnikov (Munich)Activity expansions for correlation functions: Characterizing the domain ofabsolute convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Alla Kuznetsova (Kazan)Fell bundle over a local group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Saidakhmat Lakayev (Samarkand)The threshold effects for the two-particle Hamiltonians on lattices . . . . . . . . . 26

Carlo Lucheroni (Camerino)Machine learning econometrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Alexander Lykov (Moscow)Long-time behaviour of infinite chain of harmonic oscillators . . . . . . . . . . . . . . 28

Jesper Møller (Aalborg)Determinantal point processes and their usefulness in spatial statistics . . . . . 29

Daniel Parra (Tokyo)Low-energy asymptotic in perturbed periodically twisted quantumwaveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Eugene Pechersky (Moscow)Application of the large deviations theory to a stochastic version of Hawking-Penrose black hole model. Large emission regime . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Andrey Piatnitski (Moscow)Homogenization of non-symmetric convolution type operators in periodicmedia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .33

Claude-Alain Pillet (Toulon)Thermodynamics of repeated quantum measurements . . . . . . . . . . . . . . . . . . . . . . 34

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Suren Poghosyan (Yerevan)A Characterization of the Gibbs process in terms of a given factorial measure 35

Mathias Rafler (Berlin)Integration by parts for conditioned point processes . . . . . . . . . . . . . . . . . . . . . . . . 36

Kazimierz Rajchel (Cracow)Trygonometric approach to the Schrodinger equation as a general case of knownsolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Maryam Ramezani (Bojnord)Integral equations in mathematical physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Sylvie Roelly (Potsdam)Statistical mechanical approaches for solving infinite-dimensional SDEs withmemory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Armen Shirikyan (Cergy-Pontoise)Entropy production in viscous fluid flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Hayk Sukiasyan (Yerevan)Semi-recursive algorithm of piecewise linear approximation of two-dimensionalfunction by the method of worst segment dividing . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Dimitrios Tsagkarogiannis (L’Aquila)Virial inversion and density functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Daniel Ueltschi (Warwick)Extremal states decomposition in quantum spin systems . . . . . . . . . . . . . . . . . . . 43

Nikita Vvedenskaya (Moscow)New expressions for local large deviations probability . . . . . . . . . . . . . . . . . . . . . . 44

Alexander Zass (Potsdam)Existence of Gibbsian point measures via entropy methods . . . . . . . . . . . . . . . . . 46

Elena Zhizhina (Moscow)Asymptotic pointwise estimates for heat kernels of convolution type operators 47

9

Exact solutions of a nonclassical equationwith a nonlinearity under the Laplace operator

Anatoly Aristov

Lomonosov Moscow State University, Moscowai aristov@mail.ru

In this talk, the equation

∂

∂t(∆u− u) + ∆u+ div (u∇u) = 0

is studied. Here u is a real-value function depending on the space variablex ∈ RN and time t > 0. This equation may be used to decsribe nonstationaryprocesses in liquid semiconductors.

Many books and papers are devoted to qualitative properties of solutions ofSobolev-type equations (existence, uniqueness, asymptotics), but usually suchequations are not studied in books about exact solutions.

Here, six classes of exact solutions of the named equation are constructed.They can be described analytically with some elementary and special functions.It is shown that these classes contain both bounded solutions and solutions thatgrow to infinity as time goes to a special finite value.

The method of travelling waves, the method of separating of variables andconstruction of solutions of special structures are used.

The research was supported by the Program of the President of RussianFederation for support of young PhDs (project no. MK-1829.2018.1) and par-ticularly by the grant RFBR no. 18-29-10085 mk.

References

[1] Sveshnikov A.G., Alshin A.B., Korpusov M.O., Pletner Yu.D., Linear andNonlinear Equations of Sobolev type [in Russian], Fizmatlit, Moscow, 2007

[2] Polyanin A.D., Zaytsev V.F., Zhurov A.I., Methods of Solving of NonlinearEquations of Mathematical Physics and Mechanics [in Russian], Fizmatlit,Moscow, 2005

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The phase transition for random loop models on trees

Volker Betz

TU Darmstadt, Darmstadtbetz@mathematik.tu-darmstadt.de

We show the existence of a sharp phase transition from non-existence toexistence of infinite loops for the random loop model on d-regular trees, forall dimensions d ≥ 3, and for all values of the parameter u controlling thepreference for “crosses” or “bars”. Furthermore, we give a recursive scheme toobtain an expansion of the critical parameter in powers of 1/d, which in principleis explicit but whose combinatorial complexity grows very quickly. We wereable to explicitly obtain the first 6 terms (the first two were previously foundby Ueltschi and Bjornberg by other means), and observed that (as functions ofu) they seem to have a very interesting structure.

This is a joint work with Johannes Ehlert and Benjamin Lees.

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Three-dimensional incompressible Navier-Stokes Equations: complexblow-up and related real flows

Carlo Boldrighini

La Sapienza, Romeboldrigh@mat.uniroma.it

I report results on the blow-up of a class of complex-valued solutions in-troduced by Li and Sinai and on the behavior of real solutions associated tothem.

This is a joint work with S.Frigio, D. Li, P. Maponi, A. Pellegrinotti andYa.G. Sinai.

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DLR equations and rigidity for the Sine-beta process

David Dereudre

Universite Lille 1, Lilledavid.dereudre@univ-lille1.fr

We investigate properties of Sine-beta process, the universal point processarising as the thermodynamic limit of the microscopic scale behavior in thebulk of one-dimensional log-gases, or beta-ensembles, at inverse temperaturebeta. We adopt a statistical physics perspective, and give a description ofthe Sine-beta process using the Dobrushin-Lanford-Ruelle (DLR) formalism:the restriction of Sine-beta process to a compact set, conditionally to the ex-terior configuration, reads as a Gibbs measure given by a finite log-gas in apotential generated by the exterior configuration. Moreover, we show that theSine-beta process is number-rigid and tolerant in the sense of Ghosh-Peres, i.e.the number, but not the position, of particles lying inside a compact set is adeterministic function of the exterior configuration. Our proof of the rigiditydiffers from the usual strategy and is robust enough to include more generallong range interactions in arbitrary dimension.

This is a joint work with Thomas Leble, Adrien Hardy and Mylene Maıda.

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Topology makes the spectral picture richer:quantum graph examples

Pavel Exner

Doppler Institute for Mathematical Physics and Applied Mathematics, Pragueexner@ujf.cas.cz

Spectra of periodic quantum systems are usually expected to be absolutelycontinuous, consisting of bands and gaps, the number of the latter being deter-mined by the dimensionality. If the configuration space topology is nontrivial,however, other possibilities may arise as I will illustrate using simple examples.In particular, the spectrum may then have a pure point or a fractal character,and also that it may have only a finite but nonzero number of open gaps. Fur-thermore, motivated by recent attempts to model the anomalous Hall effect, Iwill introduce a vertex coupling that violate the time reversal invariance andshow that its spectral properties are determined by the vertex degree parity.

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Regularized Hamiltonians for a three Boson system with zero-rangeinteractions

Rodolfo Figari

Complesso Universitario Monte Sant’Angelo, Napolifigari@na.infn.it

Following a suggestion of Robert Minlos in a seminal paper on the subject,I will present possible strategies to obtain lower bounded Hamiltonian for threebosons interacting through zero-range interactions.The aim is to keep the spec-tral structure of the Hamiltonians defined by Minlos at low energy (the Efimovstates) while avoiding the unboundedness from below. For this purpose I willexamine the regularizing effect of various type of ultraviolet cutoffs.

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Indirect measurements of a harmonic oscillator

Gian Michele Graf

Institut of Theoretische Physik, Zurichgian-michele.graf@itp.phys.ethz.ch

The measurement of a quantum system becomes itself a quantum-mechanicalprocess once the apparatus is internalized. That shift of perspective may resultin different physical predictions about measurement outcomes for a variety ofreasons. In fact, whereas the ideal measurement, as described by Born’s rule, isinstantaneous, the apparatus produces an outcome over time. In contrast to theoften purported view that perfect measurement emerges in the long-time limit,because decoherence supposedly improves with time, it is found that the oper-ation may be of transient character. Following an initial time interval, duringwhich the system under observation and the apparatus remain uncorrelated,there is a “window of opportunity” during which suitable observables of thetwo systems are witnesses to each other. After that time window however, theapparatus is dominated by noise, making it useless.

These conclusions are drawn from a model describing both system and ap-paratus and consisting of a harmonic oscillator coupled to a field. The equationof motion is a quantum stochastic differential equation. By solving it we es-tablish different time scales relevant to the measurement process, including aclassical and noisy large-time limit.

This is a joint work with Martin Fraas and Lisa Hanggli.

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On some inverse problems

Tigran Harutyunyan

Yerevan State University, Yerevanhartigr@yahoo.co.uk

We consider the uniqueness theorems in inverse Sturm-Liouville problems(theorems of Ambarzumian, Borg, Marchenko, Trubowitz, McLaughlin, Yurko,Horvath and others) and some their extensions. These theorems also consideredas the properties of the “Eigenvalues function of the family of Sturm-Liouvilleoperators (EVF)”, studied by author. We give the algorithm of (uniquely)reconstruction of potential by EVF. Similar problems considered for Dirac op-erators.

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Sharp phase transition for the Widom–Rowlinson model

Pierre Houdebert

University of Potsdam, Potsdampierre.houdebert@gmail.com

The Widom–Rowlinson model is formally defined as two homogeneous Pois-son point processes forbidding the points of different type to be too close. Forthis Gibbs model the question of uniqueness/ non-uniqueness depending on thetwo intensities is relevant. This model is famous because it was the first contin-uum Gibbs model for which phase transition was proven, in the symmetric caseof equal intensities large enough. But nothing was known in the non-symmetriccase, where it is conjectured that uniqueness would hold.

This is a joint work with David Dereudre.

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Phase separation and sharp large deviations

Ostap Hryniv

Durham University, Durhamostap.hryniv@durham.ac.uk

The phenomenon of “phase separation” has been at the hart of the theory ofphase transitions in low-temperature lattice systems since its discovery by Min-los and Sinai in the late 1960s. Under suitable conditions, it allows to describethe canonical ensembles of such models in terms of (families of) large contours,or “phase boundaries”, and, as a result, to study the limiting behaviour ofthe corresponding partition functions. This approach is especially successfulin two dimensions, as the resulting phase boundaries are just one-dimensionalcontours, the statistical behaviour of which is well understood.

When combined with a careful analysis of the related variational problem,these results can provide a detailed description of the typical configurationsin such ensembles. In the setting of the low-temperature Ising model on atwo-dimensional torus, the famous Dobrushin-Kotecky-Shlosman theorem rig-orously justifies the so-called Wulff construction and approximates the rescaledphase boundary by the Wulff shape, a two-dimensional region enclosed by acurve with the smallest surface energy. In turn, this implies a logarithmicasymptotic for large deviation probabilities for the total magnetisation of themodel.

In this talk, using an improved version of the DKS approach, we derive asharp large deviation principle for the total magnetisation of a low-temperatureIsing model in two dimensions.

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Cluster expansions with renormalized activities and applications tocolloids

Sabine Jansen

Ludwig-Maximilians Universitaet, Munichansen@math.lmu.de

We consider a binary system of small and large objects in the continuousspace interacting via a non-negative potential. By integrating over the smallobjects, the effective interaction between the large ones becomes multi-body.We prove convergence of the cluster expansion for the grand canonical ensembleof the effective system of large objects. To perform the combinatorial estimateof hypergraphs (due to the multi-body origin of the interaction) we exploitthe underlying structure of the original binary system. Moreover, we obtaina sufficient condition for convergence which involves the surface of the largeobjects rather than their volume. This amounts to a significant improvementin comparison to a direct application of the known cluster expansion theorems.Our result is valid for the particular case of hard spheres (colloids) for whichwe rigorously treat the depletion interaction.

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The peak model for the triplet extensions and their transformationsto the reference Hilbert space

Rytis Jursenas

Vilnius University, Vilniusrytis.jursenas@tfai.vu.lt

The classical restriction-extension theory is limited for describing supersin-gular perturbations [1] in that a symmetric restriction of a self-adjoint operatoris essentially self-adjoint in the reference Hilbert space. The typical example ofa supersingular perturbation is the point-interaction associated with a Laplaceoperator in higher dimensions. Recently [2], it has been shown that rank-foursupersingular perturbations arise in studying the formation of spin-orbit cou-pled cold molecules.

To deal with supersingular perturbations one studies triplet extensions in-stead. Here [3], we consider the so-called peak model, first initiated in [4],for the triplet extensions of supersingular perturbations in the case of a notnecessarily semibounded symmetric operator with finite defect numbers. Thetriplet extensions in scales of Hilbert spaces are described by means of abstractboundary conditions [5]. The resolvent formulas of Krein–Naimark type arepresented in terms of the γ-field and the abstract Weyl function. By applyingcertain transformation we investigate triplet extensions in the reference Hilbertspace and we show the connection with the classical extensions. In the specialcase the transformation becomes an isometric isomorphism constructed in [6].

References

[1] Kurasov P., Integr. Equ. Oper. Theory 45 (4), 2003, 437–460

[2] Jursenas R., J. Phys. A: Math. Theor. 51 (1), 2018, 015203

[3] Jursenas R., arXiv:1810.07416

[4] Kurasov P., Journal d’Analyse Mathematique 107 (1), 2009 252–286

[5] Derkach V., Hassi S., Malamud M., arXiv:1706.07948v1

[6] Langer H., Textorius B., Pacific. J. Math. 72 (1), 1977 135–165

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Direct and inverse problems in the theory of descriptionof random fields

Linda Khachatryan

Institute of Mathematics, NAS RA, Yerevanlinda@instmath.sci.am

We consider a random field on the integer lattice Zd (d ≥ 1) with the phasespaceX (X ⊂ R), namely the probability measure P on the σ-algebra generated

by all cylinder subsets of XZd

. The study of the infinite-dimensional measureP , as a rule, reduces to the study of an appropriate system P(P ) of finite-dimensional distributions generated by this measure. The main requirementfor such systems is: the system P(P ) must uniquely restore the random fieldP . The problem of construction of such system P(P ) we call a direct problemin the theory of description of random fields.

Elements of the system P(P ), being generated by the same random field,cannot be arbitrary and have to satisfy certain consistency conditions, accordingto which a system P of finite-dimensional distributions can be determinedautonomously (regardless of the random field). The problem of the existenceof a random field P such that P(P ) = P we call an inverse problem in thetheory of description of random fields.

In the theory of random processes, by the solution of the direct problem theclasses of processes are defined and their main properties are established, whilethe solution of the inverse problem makes it possible to construct models ofrandom fields with required properties. So, the standard form of the theory isthe following: first, general definition and statements, and then the constructionof models.

The situation is quite different for the Gibbs random fields. Historically,instead of being characterized by some properties of their finite-dimensional orconditional distributions (direct problem), Gibbs random fields were defined asa solution of the inverse problem for a consistent system of finite-dimensionaldistributions parameterized by infinite boundary conditions (Gibbs specifica-tion with a given potential) [1,2]. And only afterwards a purely probabilisticdefinition of a Gibbs random field in terms of its finite-dimensional distributionswas given [3,4].

In the talk, various systems of conditional distributions generated by a ran-dom field are considered: systems with finite boundary conditions, Palm-typesystem, Dobrushin-type system, and systems of one-point conditional distribu-tions with finite as well as infinite boundary conditions. We study the conditions

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under which these systems restore a random field, and also indicate their char-acteristic properties allowing to solve the inverse problem. It is shown that allthese systems are subsystems of the general system of finite-dimensional distri-butions satisfying the minimum requirements. Moreover, the elements of thegeneral system have a Gibbs form in terms of the transition energy field. Thus,the Gibbs form is universal for elements of considered systems of probabilitydistributions parameterized by boundary conditions.

This is a joint work with Boris Nahapetian.

References

[1] Dobrushin R.L., Gibbs random fields for latties systems with pair-wise in-teraction. Funct. Anal. Appl. 2, 1968, 292–301

[2] Dobrushin R.L., The problem of uniqueness of a Gibbsian random field andthe problem of phase transitions. Funct. Anal. Appl. 2, 1968, 302–312

[3] Dachian S., Nahapetian B.S., On Gibbsiannes of Random Fields. MarkovProcesses and Related Fields 15, 2009, 81–104

[4] Dachian S., Nahapetian B.S., On the relationship of energy and probabilityin models of classical statistical physics (submitted to the Journal of MarkovProcesses And Related Fields)

23

Activity expansions for correlation functions: Characterizing thedomain of absolute convergence

Leonid Kolesnikov

Ludwig-Maximilians Universitaet, Munichkolesni@math.lmu.de

In this talk, we consider Gibbs point processes with non-negative pair po-tentials (e.g., non-intersecting spheres in Rd or polymer models in Zd, but alsoincluding various models with soft-core interactions). For small activities, acluster expansion allows us to express the corresponding correlation functionsby (multivariate) power series in the activity around zero. We are primarilyinterested in the domain of absolute convergence of these activity expansions.

Those power series are given by exponential generating functions of certaincombinatorial species - in other words, their coefficients can be representedby sums over a certain class of weighted combinatorial structures. Propertiesof these combinatorial structures can be translated to structural properties ofthe corresponding generating functions - yielding the well-known Kirkwood-Salsburg relations between the activity expansions, from which one can derivea characterization of absolute convergence of the latter.

Following the so-called extended Gruber-Kunz approach introduced by Bis-sacot, Fernandez and Procacci for polymer models in Zd, we prove a necessaryand sufficient condition for absolute convergence, which is given by the existenceof a non-negative measurable function satisfying an infinite system of Kirkwood-Salsburg-type inequalities. From the latter, one can - in an elementary manner- derive well-known sufficient criteria for convergence, like the classical Kotecky-Preiss condition and the more recent Fernandez-Procacci condition (includingthe generalized versions for systems in continuous spaces with possibly soft-coreinteractions, known due to Ueltschi and Jansen).

We then present a selection of sufficient criteria for absolute convergenceobtained in some particular set-ups with hard-core interactions using this ap-proach. For non-intersecting hypercubes in Zd we derive new sufficient con-ditions, which are better than the classical Kotecky-Preiss and Gruber-Kunzconditions. In a number of other hard-core interaction systems we are able toprovide alternative proofs for known results. Some necessary conditions arisingfrom the Kirkwood-Salsburg equations for these models are briefly discussed aswell.

24

Fell bundle over a local group

Alla Kuznetsova

Kazan Federal University, Kazanalla.kuznetsova@gmail.com

Basing on the notion of local topological group we consider the “local group”G which is the local topological group with discrete topology and an additionalrequirement, namely if the elements a and b, b and c, and the elements aband c can be multiplied then the elements a and bc can be multiplied anda(bc) = (ab)c, a, b, c ∈ G. Local groups naturally arise, for instance if Γ is agroup, A is a unital C∗-algebra, π : Γ −→ A is a partial representation thenG = a ∈ Γ : π(a) 6= 0 is a local group.

We define the Fell bundle over the local group and suggest some construc-tions of C∗-algebras associated with a given Fell bundle which is a grading forrespective algebras. Also we present some examples, in particular we show theCuntz and Cuntz-Toeplitz algebras are graded over the local group.

25

The threshold effects for the two-particle Hamiltonians on lattices

Saidakhmat N. Lakaev

Samarkand State University, Samarkandslakaev@mail.ru

For a wide class of two-particle Shrodinger operators H(k) = H0(k)+V , k ∈Td being the two-particle quasi-momentum, associated to a system of twofermions on the d-dimensional lattice Zd, d ≥ 1, we prove that if the follow-ing two assumptions (i) and (ii) are satisfied, then for all nontrivial values k,

k 6= 0, the operator H(k) has eigenvalues below its threshold. The assumptions

are: (i) the two-particle operator H(0) corresponding to the zero value of thequasi-momentum has either an eigenvalue or a virtual level at the bottom of itsessential spectrum and (ii) the one-particle free Hamiltonians in the coordinaterepresentation generate positivity preserving semi-groups.

26

Machine learning econometrics

Carlo Lucheroni

University of Camerino, Camerino (MC)carlo.lucheroni@unicam.it

Many discriminative and generative machine learning methods, like neuralnetworks and hidden Markov models, have been developing in the last thirtyyears a bad reputation as to their skill in modeling and forecasting time seriesdata [1] from Finance or other research fields. This can be due to the tendencyof looking at machine learning methods as black box systems which don’t have aclear interpretation in terms of Econometrics, a discipline fully devoted to timeseries modeling and very well theoretically grounded. It is common to see papersin which batteries of different machine learning methods, from support vectormachines to recurrent networks, kernel machines, fuzzy inference systems anddecision trees are ‘tested’ and ‘compared’ in terms of their fitting or forecastingability on time series datasets, as they all were declinations of one abstractcomputational model, a Swiss Army knife, good for all needs. This black boxattitude is being even more boosted by the diffusion of easy-to-use programmingenvironments like Tensorflow/Keras that allow researchers to play with neuralnetworks by connecting their layers like Lego bricks. This approach is verylimiting, probably misleading since it leads to a wrong choice of model setups,and can be changed.

The aim of this seminar is to show that some widespread machine learningmodels do have a clear econometric interpretation. For example, recurrentneural networks can be seen as generalizations of the linear autoregressionsof the Box-Jenkins modeling frame, and univariate or vector hidden Markovmodels are actually specialized switching regime models already known fromnonlinear econometrics. Looking at these machine learning models from thiseconometric perspective can help avoid modeling mistakes when using them ontime series data, because in this case the modeler can put together a wealth ofknowledge coming from both econometrics plus discriminative modeling, andpowerful interpretation angles like generative, hidden coordinate approaches todata typical of machine learning. A few examples related to this approach willbe illustrated for Energy Finance time series.

References

[1] Makridakis S., Spiliotis E., Assimakopoulos V., Statistical and MachineLearning forecasting methods: Concerns and ways forward, PLoS ONE 13(3), e0194889, 2018. DOI: 10.1371/journal. pone.0194889

27

Long-time behaviour of infinite chain of harmonic oscillators

Alexander Lykov

Lomonosov Moscow State University, Moscowalekslyk@yandex.ru

We consider infinite number of point particles ... < xk < xk+1 < ..., k ∈ Z,on R (infinite chain of oscillators) with formal Hamiltonian

H =∑k

v2k

2+ω2

0

2

∑k

(xk − ka)2 +ω2

1

2

∑k

(xk+1 − xk − a)2, a > 0,

y = yk(t) = xk(t)− ka, v(t) = yk = xk, M(t) = supk∈Z|yk(t)|.

and present some results concerning stability (in l∞) of its fixed point (zeroenergy point) with respect to various perturbations.

Theorem (Theorem 1.). Suppose that y(0), v(0) ∈ l2(Z).1. If ω0 > 0, then supt>0M(t) <∞.2. If ω0 = 0 then for all t > 0 the following inequality holds:

M(t) 62√ω1||v(0)||2

√t+ ||y(0)||2

but for all δ > 1/2 there is initial condition y(0) = 0, v(0) ∈ l2(Z) such that

limt→∞y0(t)√t

lnδ t = Γ(δ) > 0 (Γ is the gamma function).

Theorem (Theorem 2.). If ω0 = 0 and v(0) = 0 then:1. If y(0) ∈ l∞(Z), then for all t > 0, M(t) 6

(c√t+ 2

)M(0) for some con-

stant c > 0.2. If yk(0), k ∈ Z is i.i.d and bounded in k with probability one (i.e. supk |yk(0)| <∞ a.s.) then for all n ∈ Z

P(lim supt→∞

yn(t) = +∞) = P(lim inft→∞

yn(t) = −∞) = 1.

This is a joint work with Vadim Malyshev and Margarita Melikian.

References

[1] Lykov A.A., Malyshev V.A., From The N-Body Problem to Euler Equations,Russian Journal of Mathematical Physics 24, 2017, 79–95

[2] Lykov A.A., Malyshev V.A., Melikian M.V., Phase diagram for one-waytraffic flow with local control, Physica A: Statistical Mechanics and its Ap-plications 486, 2017, 849–866

28

Determinantal point processes and their usefulness in spatial statistics

Jesper Møller

Aalborg University, Aalborgjm@math.aau.dk

We discuss how the appealing properties of determinantal point processes(DPPs) can be used for constructing statistical models and methods, exploitingthat the likelihood and moment expressions can be evaluated and realizationscan be simulated in a simple way, [1–4], where freely available software havebeen developed for DPPs in space or on the sphere. In this talk we pay par-ticular attention to DPPs in space and consider to some extent DPPs on thesphere. Following [5] we also discuss how to quantify repulsiveness in a generalsetting based on a simple coupling result for a DPP and any of its reduced Palmdistributions.

References

[1] Lavancier F., Møller J., Rubak E., Determinantal point process models andstatistical inference, Journal of Royal Statistical Society: Series B (Statis-tical Methodology) 77, 2015, 853–877

[2] Lavancier F., Møller J., Rubak E., Determinantal point process models andstatistical inference: Extended version, arXiv:1205.4818, 2014

[3] Møller J., Nielsen M., Porcu E., Rubak E., Determinantal point processmodels on the sphere, Bernoulli 24, 2018, 1171–1201

[4] Møller J., Rubak E., Functional summary statistics on the sphere with anapplication to determinantal point processes, Spatial Statistics 18, 2016,4–23

[5] Møller J., O’Reilly E., Couplings for determinantal point processes andtheir reduced Palm distributions with a view to quantifying repulsiveness,(submitted for journal publication), arXiv:1806.07347, 2018

29

Low-energy asymptotic in perturbedperiodically twisted quantum waveguides

Daniel Parra

University of Tokyo, Tokyodparra@ms.u-tokyo.ac.jp

In this talk, we consider the Dirichlet Laplacian in a three-dimensionalwaveguide obtained as a perturbation of a periodically twisted tube. The per-turbation consists of both bending and twisting and depends on a couplingparameter. We expand the resolvent of the perturbed operator near the bot-tom of its essential spectrum, we show the existence of exactly one resonancein the asymptotic regime, and we compute the leading term of the asymptotic.

30

Application of the large deviations theory to a stochastic version ofHawking-Penrose black hole model. Large emission regime

Eugene Pechersky

Institute for Information Transmission Problems of RAS, Moscowpech@iitp.ru

We consider a piece wise constant Markov process ξ(t) on finite interval[0, T ]. The work devoted to an application of the large deviations theory to theconsidered Markov process. The process is a version of a stochastic dynamicsthat we have imposed on the simplest model of the black hole introduced by S.Hawking and R. Penrose [1,2]. The model consists of a volume and N particles(photons) in the volume. A part of the volume forms the black hole. Theboundary of the black hole is called black hole horizon. The particles are splitinto two groups: one is located in the black hole interior another outside of theblack hole, that is in the black hole exterior. The stochastic dynamics describestransitions of the particles from the black hole to the outside, the (emission),and from the outside to the interior, the (absorption).

The value ξ(t) of the Markov process is the number of the particles in theblack hole interior at the time moment t. The stochastic dynamics is definedby intensities of the jumps. Let ξ(t) = k. The intensity of the absorptionk → k+1 (k < N) is proportional to k2 that means proportionality to the blackhole horizon. The intensity of the emission k → k − 1 (k > 0) is proportionalto 1

k2 that means inverse proportionality to the black hole horizon.The main interest of our studies is a functional η(t) of the process path ξ(t)

which means the number of the emissions on the interval [0, t], t ≤ T . Westudy the probability of the large emission number on the interval [0, T ]. Thecorresponding event is (η(T ) > BT ), where B > 0 is large. We use the largedeviations theory for this study to obtain the asymptotic of the logarithm ofthe probability of this event or events close to this one. The asymptotic isconsidered as N →∞. For the mentioned event we have to evaluate

limN→∞

1

Nln Pr(η(T ) > NBT ). (1)

For application the large deviations theory we find the rate function

I(x,y) =

∫ T0

supκ1(t),κ2(t)

κ1(t)x(t) + κ2(t)y(t)− λx2(t)(1− x(t))[eκ1(t) − 1]

− µ1

x2(t)[e−κ1(t)+κ2(t) − 1]

dt,

31

where x(·) and y(·) are differentiable functions on [0, T ]. The function x(·)takes its values in the interval [0, 1] and means the density of the particles inthe black hole. The function y(·) takes its values in R+, is non-decreasing andmeans the intensities of the emissions. The rate function is Legendre transformof Hamiltonian

H(x, y,κ1,κ2) = λx2(1− x)[eκ1 − 1] + µ1

x2[e−κ1+κ2 − 1].

Knowing the rate function I we can find the optimal path corresponding thestudied amount of the emission. To this end we seek the infimum of the ratefunction over a certain set of the paths. For example, for the emission in (1)we have to find inf I(x,y), where the infimum is taken over all paths (x,y)such that y(0) = 0,y(T ) = BT . The problem is reduced to solving the Hamil-tonian system of equations corresponding to Hamiltonian H with the boundaryconditions y(0) = 0,y(T ) = BT .

In the considered case the Hamiltonian system is highly nonlinear. It seemsto find a solution of this boundary problem is very problematical. Therefore wediscuss several cases of other boundary problems for these equations giving ananswer for events included in the mentioned.

This is a joint work with S. Pirogov and A. Yambartsev.

References

[1] Hawking, S.W., Breakdown of predictability in gravitational collapse, Phys.Rev. 14 (10), 1976, 2460–2473

[2] Penrose, R., Singularity and time-asymmetry, in General relativity: an Ein-stein centenary survey, eds Israel, Hawking, Cambridge, 1979, 581–638

32

Homogenization of non-symmetric convolution type operatorsin periodic media

Andrey Piatnitski

Arctic University of Norway, campus Narvik,Institute for Information Transmission Problems of RAS, Moscow

apiatni@iitp.ru

The talk will focus on a homogenization problem for a nonlocal convolutiontype not necessary symmetric operators of the form

Au(x) =

∫Rd

a(x− y)Λ(x, y)(u(y)− u(x)

)dy

with a periodic coefficient Λ(x, y). We assume that• a(z) ≥ 0,

∫Rd a(z) dz = 1,

∫Rd |z|2a(z) dz,< +∞,

• There exist λ− > 0 and λ+ > 0 such that λ− ≤ Λ(x, y) ≤ λ+.Under these conditions the operator A is the generator of a Markov semigroupin L2(Rd). In order to characterize the large time behaviour of this semigroupwe make a diffusive scaling with a small positive parameter ε > 0 and arrive atthe following family of operators:

Aεu(x) =1

εd+2

∫Rd

a(x−yε

)Λ(xε ,

yε

)(u(y)− u(x)

)dy.

Our goal is to study the limit behaviour of this family as ε→ 0.One can show that for any ε > 0 and any m > 0 the operator Aε − m is

invertible in L2(Rd), that is an equation (Aε −m)u = f has a unique solutionuε ∈ L2(Rd) for any f ∈ L2(Rd).

Theorem. There exists a constant vector b ∈ Rd and an elliptic operatorAeffu = aij

∂2

∂xi∂xju with a constant symmetric positive definite matrix a such

that for any f ∈ L2(Rd) and m > 0 the solution uε of the equation (Aε −m)u = f converges, as ε→ 0, in L2(Rd) to a solution of the effective equationAeffu−mu = f .

This is a joint work with Elena Zhizhina.

33

Thermodynamics of repeated quantum measurements

Claude-Alain Pillet

Universite de Toulon, Toulon,Editor-in-Chief of Annales Henri Poincare

pillet@univ-tln.fr

Under appropriate circumstances, subjecting a quantum system to repeatedmeasurements with possible outcomes in a finite set A generates an invariantmeasure P of a simple classical dynamical system: the left shift

τ : (ω1, ω2, ω3, . . .) 7→ (ω2, ω3, ω4 . . .)

on the set AN. However, except in very special cases, the resulting dynamicalsystem (AN, τ,P) does not belong to well studied classes. Viewed as one-dimensional spin systems with long range interactions, these systems exhibitvery rich (and sometimes very singular) thermodynamic behaviour. We willoutline a general thermodynamic formalism which accommodates these systemsand illustrate its unexpected features on a number of examples.

This talk is based on joint works with T. Benoist, N. Cuneo, V. Jaksic andY. Pautrat.

34

A Characterization of the Gibbs process in termsof a given factorial measure

Suren Poghosyan

Institute of Mathematics, NAS RA, Yerevansuren.poghosyan@unicam.it

Given a reference Radon measure %z, indexed by some non negative functionz, a pair potential Φ, satisfying Penrose stability, regularity and local integra-bility, both defined on some locally compact second countable Hausdorff spaceX, we define a function which is shown to be the correlation function of a pro-cess Pz. Under these conditions Pz is uniquely determined by its correlationfunction.

Under more restrictive conditions on (Φ, %z), we show that for sufficientlysmall z the set G(Φ, %z) of Gibbs processes for Φ is either empty or a singletonconsisting of the process Pz.

This is a joint work with Hans Zessin.

35

Integration by parts for conditioned point processes

Mathias Rafler

Technical University of Berlin, Berlinafler@math.tu-berlin.de

Integration by parts formulas characterize Papangelou processes as reversiblelaws of certain spatial birth-and-death processes. Typically, conditioning thesepoint processes on events of probability zero, e.g. by fixing the barycentre orthe first moment of the random point configurations, is not compatible withthese dynamics. The method shown in Conforti, Kosenkova and Roelly to de-rive an integration-by-parts formula for a class of Poisson processes is extendedto a larger class of Papangelou processes.

36

Trygonometric approach to the Schrodinger equationas a general case of known solutions

Kazimierz Rajchel

Institute of Computer Science, Pedagogical University of Cracow, Cracowkazimierz.rajchel@up.krakow.pl

General solutions of the Schrodinger equation were obtained by means ofparametrization of the unit circle equation, related to the Riccati equation.Particular solutions, such as oscillator, Coulomb or Morse potentials, are givenby appropriate choice of parameters. This concept is strictly connected withthe theory of orthogonal polynomials and hypergeometric functions. As a resultone can choose more objectively orthogonal basis in numerical computations,which leads to improvement of accuracy.

37

Integral equations in mathematical physics

Maryam Ramezani

University of Bojnord, Bojnordm.ramezani@ub.ac.ir

Integral equation is encountered in a variety of applications in many fieldsincluding continuum mechanics, quantum mechanics, geophysics, electricity andmagnetism, kinetic theory of gases, hereditary phenomena in physics and biol-ogy, and etc.

Our purpose here is to study the existence of solutions for the following heatintegral equation in the space Lp where 0 < p ≤ 1.

u(x, t) =

∫ t

0

∫ x

x+t−1−τk(x− ξ, t− τ) F (ξ, τ, u(ξ, τ)) dξ dτ (1)

where x ∈ R, t ∈ I = [0, 1] and k(x, t) = exp

(x− tx+ t

).

Let Lp = f : [0, 1]→ R : f is measurable, ρ(f) =∫ 1

0 |f(x)|pdx <∞.To study the integral equation (1) we set the following hypotheses:(H1) F : R× I ×B → B is a ρ-continuous function.(H2) F (x, t, u) ≥ F (x, t, v) for each u, v ∈ B with u ≥ v a.e. and for all

(x, t) ∈ R× I.(H3) ρ(F (x, t, u) − F (x, t, v)) ≤ ρ(u − v) for each u, v ∈ B with u ≥ v a.e.

and for all (x, t) ∈ R× I.

Theorem. Let the above conditions are satisfied. Then the integral equation (1)has a unique solution in X.

This is a joint work with H. Ramezani.

38

Statistical mechanical approaches for solvinginfinite-dimensional SDEs with memory

Sylvie Roelly

Institute of Mathematics, University of Potsdam, Potsdamroelly@math.uni-potsdam.de

We present existence (and uniqueness) results on weak solutions of infinite-dimensional stochastic differential equations driven by a Brownian term, ob-tained in collaboration over the years with P. Dai Pra, W. Ruszel and D.Dereudre. The main object under study is the infinite-dimensional Stochas-tic Differential Equation

dXi(t) = b(t, θiX) dt+ dBi(t), i ∈ Zd,

on the configuration space Ω = C([0, T ],R)Zd

where the drift b : [0, T ]×Ω is anadapted functional, θi denotes the space-shift on Ω by vector −i and (Bi)i∈Zd isa sequence of independent real-valued Brownian motions. Our aim is to provethe existence/uniqueness of a space-shift invariant weak solution of the aboveSDE on any finite time-interval [0, T ] where the drift b is supposed to be asgeneral as possible, in particular with memory, non-regular and non-bounded.We only suppose that it is local and admits a sublinear growth. Delay equationswith non continuous drift are typical examples.

A fruitful approach to construct weak solutions of infinite random systemsis to describe them as Gibbs measures on a path space. Here, we make useof an entropy method (going back to Georgii) to solve the existence problemof limit points of finite volume solutions. Then we identify the infinite volumelimit measures as Gibbs measure via a variational problem.

Our approach underlines to what extent tools from Statistical Mechanicscan be powerful in the framework of Stochastic Analysis.

39

Entropy production in viscous fluid flows

Armen Shirikyan

University of Cergy-Pontoise, Cergy-Pontoisearmen.shirikyan@u-cergy.fr

The concept of entropy production in non-equilibrium statistical mechanicsis related to the phenomenon of irreversibility. We shall use a simple exampleof a finite-state Markov chain to illustrate the main objects and some typicalresults, including the law of large numbers, the central limit theorem, largedeviations, and the fluctuation relation. We next turn to the motion of a particlein a viscous fluid flow described by the 2D Navier–Stokes equations. We discussrecent achievements for this model and formulate some open questions.

This is a joint work with V. Jaksic, V. Nersesyan, and C.-A. Pillet.

40

Semi-recursive algorithm of piecewise linear approximation oftwo-dimensional function by the method of worst segment dividing

Hayk S. Sukiasyan

Engineering University of Armenia; Institute of Mathematics, NAS RA,Yerevan

haik@instmath.sci.am

In the numerical solution of two-dimensional non-linear boundary valueproblems of mathematical physics, the finite element method is often used.This method assumes that the domain of the boundary problem is divided intosmall sub-domains (elements) within which the desired function is assumed tobe linear. Thus, the desired function is approximated by a piecewise linearfunction. Its graph consists of triangles, the projections of which on the OXYplane form a triangular mesh.

In recent years, meshes with variable number of nodes are often used, i.e.the process of successive approximations extends not only to the approximatedfunction, but also to the corresponding grid. At the same time, additionalnodes are sequentially added in the worst (in terms of approximation error)sub-domains. Thus, the mesh is successive improved and the approximationerror is minimized. For practical reasons, it is preferable to recursive algorithmsof mesh generation, which, when a new vertex is added, the previous verticesand edges remain in place.

We have constructed a semi-recursive algorithm for constructing a piecewiselinear approximation of a two-dimensional function by dividing the worst seg-ment. When adding a new vertex, all previous vertices and almost all edgesremain in their places. The edge may change if the ”flip” operation is applicableto it: replacing a longer diagonal with a shorter one in a tetragon.

Theorem. For any approximated function and any number of vertices, the meshresulting from the operation of the semi-recursive algorithm is a Delaunay tri-angulation.

The developed approach is implemented on a model problem of approxima-tion of a two-dimensional parabolic function.

This is a joint work with Tatev Melkonyan.

41

Virial inversion and density functionals

Dimitrios Tsagkarogiannis

University of L’Aquila, L’Aquiladimitrios.tsagkarogiannis@univaq.it

Deriving equations of state that relate thermodynamic quantities is one ofthe main challenges of both theoretical and computational methods in statis-tical mechanics. One key rigorous result in this direction was the proof of theconvergence of the virial expansion by Lebowitz and Penrose in 1964. The mainapproach in that as well as in subsequent works was to express the activity as afunction of the density and substitute it in the pressure expansion. The proofconsists of proving that the composed map is absolutely convergent. Puttingtogether this result with previous ones which identify the new coefficients of thedensity expansion with the two-connected (irreducible) graphs one also obtainsthe convergence of the latter power series expansion (i.e., of the composed mapbeing a graph re-summation). A similar approach can be used in order to ex-pand the truncated correlation function with respect to the density. However,for other expansions such as the direct correlation function with respect to thedensity or the free energy with respect of the truncated correlation function theproof of convergence of the corresponding re-summation is more delicate (eventhough the virial expansion it is well understood).

Such expansions have been developed in the 60’s mainly by the works ofHiroike-Morita and of Stell and gave rise to a number of closures and approx-imate methods extensively used until nowadays in liquid state calculations. Acommon key step in these approaches was the replacement of chunks of thegraphs in the activity by graphs over the density (or the truncated correlation).Furthermore, the general framework in these works is the inhomogeneous gas,namely densities and activities which depend on the position, while the con-vergence proof of Lebowitz and Penrose as well as of all subsequent papers wasvalid in the homogeneous case.

The goal of this talk, after giving a short description of the above, is two-fold:1. Establish the validity of the inversion formulas for the case of inhomogeneousfluids. 2. Prove the validity of the re-summation operations.

The key technical tool is a combinatorial identity for trees which also givesan improvement of the radius of convergence for the (homogeneous) virial ex-pansion as first established by Lebowitz and Penrose. Moreover, working in theinhomogeneous set-up, applications include systems in classical density func-tion theory, liquid crystals as well as molecules with various shapes (internaldegrees of freedom). This is joint work with Sabine Jansen and Tobias Kuna.

42

Extremal states decomposition in quantum spin systems

Daniel Ueltschi

University of Warwick, Warwickueltschid@gmail.com

In equilibrium statistical physics, the set of infintie-volume Gibbs states hasthe structure of a Choquet simplex, that is, each state can be written as a convexcombination of extremal states. This is known very generally and abstractly.The challenge is to identify the extremal states in given models. Followinga suggestion by Tom Spencer, we consider a sort of Laplace transform of themeasure on extremal states. We can calculate it for spin 1/2 Heisenberg modelsand for the quantum interchange model on the complete graph.

This is a joint work with Jakob Bjornberg and Jurg Frohlich.

43

New expressions for local large deviations probability

Nikita Vvedenskaya

Institute for Information Transmission Problems of the Russian Academy ofSciences, Moscow

ndv@iitp.ru

We consider a continuous-time Markov process ξ(t) on Z+ with nearest-neighbor jumps, the intensity η(x) = λ(x) + µ(x) and jump rates

λ(n)

η(n)for the birth-jump n→ n+ 1,

µ(n+ 1)

η(n)for the death-jump n+ 1→ n, n ∈ Z+.

The asymptotical behavior of functions λ(n) and µ(n) is described by

limx→∞

λ(x)

T blxly(x)= lim

x→∞

µ(x)

T bmxmz(x)= 1

where b, l,m ∈ (0,∞), max(l,m) > 0, and y(x), z(x) are slowly varying func-tions. (A processes of this type can explode in a finite time.)

Given T > 1, we consider the scaled process ξb,1(t). Here ξb,1(t) =ξ(Tt)

T b,

0 ≤ t ≤ τ ∧ 1, Tτ being the time of explosion (finite or infinite).Fix a function x ∈ (0, 1] 7→ f(x) > 0 of class C1, with f(0) = 0. Given

ε > 0, let Uε(f) denote the set of cadl‘ag functions g such that |f(t)− g(t)| ≤ ε∀ t ∈ [0, 1]. We are interested in the exponential asymptotic of probabilityP(τ > 1, ξb,1( · ) ∈ Uε(f)), i.e., in the limiting relations

limε→0

lim supT→∞

log P(τ > 1, ξb,1( · ) ∈ Uε(f))

ψ(T )=

= limε→0

lim infT→∞

log P(τ > 1, ξb,1( · ) ∈ Uε(f))

ψ(T )= −I(f). (1)

Theorem. [1] Assume that l 6= m. Then the equations (1) hold true. Moreover,

I(f) =∫ 1

0 [f(t)](max l,m)dt and ψ(T ) = max[T bly(T ), T bmz(T )] .

Theorem. [2] Assume that l = m and limx→∞y(x)

z(x)6= 1. Then the equations

(1) hold true, I(f) =∫ 1

0 [f(t)]ldt,

44

ψ(T ) = T bl+1(√y(T )−

√z(T ))2 as T b < T bl+1 + T (bl+1)/2,

ψ(T ) = T bl+1(y(T ) + z(T )) as T b > T bl+1 + T (bl+1)/2.

We use the approach developed in [1]. The presented theorems generalizethe results of [2].

References

[1] Mogulskii A., Pechersky E., Yambartsev A., Large deviations for excursionsof non-homogeneous Markov processes, Electronic Commun. Probab. 19,2014, 1–8

[2] Vvedenskaya N.D., Ljgachov A.V., Suhov Y.M., Yambartsev .A.A., A LocalLarge Deviation Principe for a Class of Birth-Death Processes, InfomationTransm. Probl. 54 (3), 2018, 263–280

45

Existence of Gibbsian point measures via entropy methods

Alexander Zass

University of Potsdam, Potsdamzass@math.uni-potsdam.de

Existence of infinite-volume Gibbs measures can be proved using specificentropy as a Lyapunov function, as proposed by Georgii. Drawing inspirationfrom examples in stochastic geometry, we use this approach in the framework ofmarked Gibbs point processes and present an existence result for infinite range(and unbounded) interactions.

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Asymptotic pointwise estimates for heat kernelsof convolution type operators

Elena Zhizhina

Institute for Information Transmission Problems, Moscowejj@iitp.ru

We consider asymptotic estimates of the heat kernel (=fundamental solu-tion) of certain evolution equations with non-local elliptic part:

∂tu− Au = 0,

where A is a non-local operator given by

Au = a ∗ u− u. (1)

The convolution kernel a is such that

a(x) ≥ 0; a(x) = a(−x); a(x) ∈ L∞(Rd) ∩ L1(Rd),∫Rd

a(x)dx = 1,

∫Rd

|x|2a(x)dx <∞.

In addition we assume that convolution kernels a(x) decay at infinity at leastexponentially and admit an estimate from above by a radially symmetric func-tion: a(x) ≤ ce−b|x|

p

with b > 0 and p ≥ 1.The large time behaviour of the studied heat kernel depends crucially on

the relation between |x| and t. We consider separately four different regions in(x, t) space, namely,

(i) |x| ∼ t1/2, (ii) t12 |x| t, (iii) |x| ∼ t, (iv) |x| t.

Remark that the fundamental solution is the same as the transition densityfor the corresponding continuous time jump process in continuum with indepen-dent increments generated by the operator (1), and the region (i) correspondsto the standard deviations where the local central limit theorem applies, (ii) isthe region of the moderate deviations, (iii) is the region of large deviation, and(iv) should probably be called the “extra large” deviation region.

We compare the obtained asymptotics with the heat kernel of the classicalheat equation ∂tu−∆u = 0, which is given by the Gauss-Weierstrass function.

References

[1] Grigoryan A., Kondratiev Yu., Piatnitski A., Zhizhina E., Pointwise esti-mates for heat kernels of convolution type operators, Proc. London Math.Soc. 117 (4), 2018, 849–880

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