Ian Davies Recent work has seen a return to Poisson-Lévy excursion measures with a paper documenting the full construction of the next to leading order term in the (asymptotic) expansion of the Poisson-Lévy excursion measure [1]. It transpires that this term is identically zero and so we have the leading order term (having one of only two possible forms) giving an excellent approximation. One may extend the result, if sufficiently patient, as all necessary structures and techniques are discussed. A number of collaborations with colleagues in the School of Engineering, Prof Y Feng, Dr C Li and Prof R Owen FRS, have centered on the development of a mathematically rigorous theory of continuum mechanics in stationary stochastic fields, a prototype analysis system for elasticity problems of random media. A Fourier-Karhunen-Loève discretization scheme is developed which exhibits a number of advantages over the widely used Karhunen-Loève expansion scheme based on Finite Element meshes, including better computational efficiency in terms of memory and CPU time, a convenient a priori error-control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Two papers have appeared [2, 3] with other short communications in conference proceedings. My work, with Aubrey Truman and Huaizhong Zhao, concerning the singularities of the stochastic heat and Burgers equations has been most fruitful. We have shown that a knowledge and deep understanding of the stochastic Hamilton principal function allows one to determine the nature of the caustics for the inviscid Burgers equation and the corresponding “wavefronts” for the stochastic heat equation. There are two preprints, and they may be downloaded from the Texas Mathematical Physics archive [4, 5]. Our first substantial paper [6] in this area gives the detailed exposition for the geometrical properties. It appeared in under the title ‘Stochastic heat and Burgers equations and their singularities I, Geometrical properties’. Our second paper [7] appeared under the title ‘Stochastic heat and Burgers equations and their singularities II. Analytical properties and limiting distributions’. Selected Publications [1] Electronic Journal of Probability, 13, (2008), 1283 – 1306 [2] International Journal for Numerical Methods in Engineering, 73, (2008), 1942 – 1965 [3] Engineering Computations, 23, (2006), 794 – 817 [4 ] Stochastic Heat and Burgers Equation, 01 – 45 [5] Stochastic Heat and Burgers Equation and their singularities II, 04 – 283

[6] Journal of Mathematical Physics 43, (2002), 3293 – 3328 [7] Journal of Mathematical Physics 46, (2005), no.4, 043515-1 – 043515-31

Kristian Evans My research topic is based around the analysis of jump Markov processes, in particular by looking at the generators of these processes. These generators are pseudo-differential operators the symbols of which are continuous negative definite functions (in the sense of Schoenberg). I started my research career by looking at variable order subordination, i.e. by looking at subordinate generators using state space dependent Bernstein functions. More recently I have undertaken research into pseudo-differential operators acting on functions defined on $R^n \times Z^m$. The aim is to construct stochastic processes which in some part of the state space will behave like Feller processes on $R^n$, while in other parts they behave like Markov chains. The approach starts with symbols defined on $R^n \times Z^m\times R^n\times T^m$ and to prove for corresponding pseudo-differential operators a priori estimates allowing us to eventually apply the Hille-Yosida theorem to construct a corresponding positivity preserving semigroup and hence a process with state space $R^n \times Z^m$. Publications [1] Evans, K.P., and N. Jacob, Feller Semigroups Obtained by Variable Order Subordination. Rev. Mat. Complut. 20 (2007), 293-307. Erratum: Rev. Mat Complut. 22 (2009), 303–304 [2] Evans, K.P., and N. Jacob, Variable Order Subordination in the Sense of Bochner and Pseudo-Differential Operators. Math. Nachr. 284(2011), 987–1002. [3] Evans, K.P., and N. Jacob, Q-matrices as Pseudo-Differential Operators with Negative Definite Symbols. Math. Nach. 286(2013), 631–640. [4] Evans, K.P, O. Morris, and N. Jacob. On a class of Pseud-Differential Operators in $R^n \times Z^m$ generating Feller Semigroups (in press)

Dmitri Finkelshtein My research interests and experience concern different topics of functional analysis, probability theory, applications in physics, biology, and economics. (1) Models inspired by sciences. Complex systems theory is a quickly growing interdisciplinary area with a very broad spectrum of motivations and applications. In consideration of biological applications, complex adaptive systems are characterized by such properties as diversity and individuality of

components, localized interactions among components, and the outcomes of interactions used for replication or enhancement of components. In the study of these systems, proper language and techniques are delivered by the interacting particle models which form a rich and powerful direction in modern stochastic and infinite dimensional analysis. Interacting particle systems are widely used as models in condensed matter physics, chemical kinetics, population biology, ecology (individual based models), and sociology and economics (agent based models). In [10], we developed a rigorous description and studied properties of the so-called Bolker–Pacala–Dieckmann–Law model in spatial ecology. We continue these investigations in [1]. In [3], we also rigorously described an influence of establishment and fecundity in the evolution on the microscopic level. In [9], various regulation mechanisms for a free development economic model were studied too. (2) Stochastic dynamics of interacting particle systems in continuum. Systems of interacting particle systems in continuum give rigorous mathematical description for real systems with huge number of elements, like for example systems of molecules in mathematical physics, plants or insects in mathematical biology, agents in mathematical economics. The evolution of such systems may be described as Markov stochastic dynamics, in particular, birth-and-death dynamics and diffusion dynamics (discrete or continuous). A mathematical description for the corresponding evolutions of states in terms of dynamics of correlation functions, quasi-observables and generating functionals for systems with one or several types of elements was given. We studied also different types of specific birth-and-death and conservative stochastic dynamics. In particular, in [4], an approximative scheme and construction of a non-equilibrium Glauber dynamics in continuum were developed and the corresponding ergodic properties were studied. In [5], a semigroup approach to construction of non-equilibrium dynamics for general birth-and-death stochastic evolutions was realized. (3) Multiscale descriptions of complex systems. Study of interacting particle systems may be realized in different levels of scales. The microscopic description deals with Markov generators in state spaces and the corresponding realizations in terms of correlation functions mentioned above. Another level is provided by the mesoscopic description which yields comprehension of the inner properties of the system under consideration. We proposed a general scheme of a mesoscopic scaling procedure for birth-and-death and conservative systems. In [2,3,5,6] this scheme was realized at a rigorous level for some specific models. In particular, sufficient conditions for convergence of rescaled hierarchy of correlation functions to a limiting hierarchy were found. The limiting hierarchies preserve chaotic properties of initial state and lead to a new class of kinetic-type equations, which are nonlocal and nonlinear equations. For a number of such specific equations we proved the existence and uniqueness of their solutions in spaces of bounded nonnegative functions. In particular, in [7], for a non-equilibrium dynamics of binary jumps the corresponding Boltzmann-type integro-differential equation was studied. Another equation which appeared in course of scaling of the Glauber evolution is the dynamical version of the well-known Kirkwood--Monroe equation from freezing theory, see [6].

(4) Analysis on spaces of configurations. Spaces of locally finite subsets (configurations) of an underlying space play an important role in modern infinite-dimensional analysis. In particular, properties of Laplace operator on configuration space over a domain with boundary and the corresponding spectral problems were considered. Measures on configuration spaces (e.g., Poisson, Gibbs and others) play a fundamental role for statistical description of stochastic evolutions and they are also important objects for theoretical considerations, see e.g. [7]. Measures on configuration spaces defined by relative energies were studied, in particular, the corresponding existence and uniqueness problems were discovered. These results were extended to the case with configurations of elements of different types. Selected Publications [1] O. Ovaskainen, D. Finkelshtein, O. Kutovyi, S. Cornell, B. Bolker, Yu. Kondratiev. A mathematical framework for the analysis of spatial-temporal point processes. Theoretical Ecology, (2013), doi: 10.1007/s12080-013-0202-8. [2] D. Finkelshtein, Y. Kondratiev, and Y. Kozitsky. Glauber dynamics in continuum: a constructive approach to evolution of states. Discrete and Cont. Dynam. Syst.-Ser A. 33(4), (2013), 1431–1450. [3] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Establishment and fecundity in spatial ecological models: statistical approach and kinetic equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 16(2) (2013), 1350014 (24 pages). [4] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Correlation functions evolution for the Glauber dynamics in continuum. Semigroup Forum 85 (2012), 289–306. [5] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Semigroup approach to non-equilibrium birth-and-death stochastic dynamics in continuum. J. of Funct. Anal. 262(3) (2012), 1274–1308. [6] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Vlasov scaling for the Glauber dynamics in continuum. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 14(4) (2011), 537–569. [7] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binary jumps in continuum. I. Equilibrium processes and their scaling limits. J. Math. Phys. 52 (2011), 063304:1–25. [8] D. Finkelshtein, Y. Kondratiev, O. Kutoviy, and E. Lytvynov. Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit. J. Math. Phys. 52 (2011), 113301:1–27. [9] D. Finkelshtein and Y. Kondratiev. Regulation mechanisms in spatial stochastic development models. J. Stat. Phys. 136(1) (2009), 103–115.

[10] D. Finkelshtein, Y. Kondratiev, and O. Kutoviy. Individual based model with competition in spatial ecology. SIAM J. Math. Anal. 41(1) (2009), 297–317.

Niels Jacob Jacob has established a new approach to jump-type Feller processes and Dirichlet processes using pseudo-differential operators, the symbols of which are the analogue to characteristic exponents of Lévy processes. His and his students’ achievements until ca. 10 years ago are documented in the monograph “Pseudo-differential Operators and Markov Processes, Vol. 1 – 3” , published 2001 – 2005, [1] – [3]. Meanwhile he pioneered a geometric approach for the understanding of the transition densities of these processes. First results are published jointly with former PhD-students [4], as well as [5] which extents some of these considerations to Markov chains. Jacob is frequently invited as to international conferences, for example SPA 2013 at Boulder, as he is often serving on the scientific committees of international conferences, for example “Stochastic Processes, Analysis and Mathematical Physics” Osaka 2014, which is an ICM2014 satellite conference. So far he has supervised 17 PhD-students, currently 3 PhD-students are working with him. Selected Publications [1] Jacob, N., Pseudo-differential Operators and Markov Processes. Vol. 1: Fourier Analysis and Semi-groups. Imperial College Press, 2001. [2] Jacob, N., Pseudo-differential Operators and Markov Processes. Vol. 2: Generators and their Potential Theory. Imperial College Press, 2002. [3] Pseudo-differential Operators and Markov Processes. Vol. 3: Markov Processes and Applications. Imperial College Press, 2005. [4] Jacob, N., Knopova, V., Landwehr, S., Schilling, R., A Geometric Interpretation of the Transition Density of a Symmetric Lévy Process. Science China. Mathematics. 55 (2012), 1099 – 1126. [5] Evans, K., Jacob, N., Q-matrices as Pseudo-differential Operators with Negative Definite Symbols. Math. Nachr. 286 (2013), 631 – 640.

Mark Kelbert To date Kelbert published over 100 peer-reviewed papers and 8 books. (1) Absence of continuous symmetry-breaking in 2D quantum systems. Symmetry-breaking types of quantum phase transitions are important in solid-state

physics, optics and physics of new materials. The latter discipline is focusing on unusual properties of materials like graphene which is an extremely thin layer of carbon, transparent for light but impregnable for gases or liquids. Materials of this type are essentially 2D; they are believed to have the big future (the Nobel Prize in Physics in 2010 was awarded for theoretical and experimental works on graphene). Physicists predicted for a long time that, e.g., if a Hamiltonian is invariant under an action of a Lie group then every Gibbs state should also be invariant under such an action. In a popular folklore this is called the Mermin--Wagner theorem. There exists notable mathematical literature where such statements have been rigorously proven for classical systems with continuous systems in two dimensions. However, rigorous proofs cover two-dimensional quantum systems rather poorly or do not cover them at all. These results are rigorously proved in [4] for quantum Hamiltonian. They are extended to the Hubbard model of quantum transport in [2]. The basic tool will be the Feynman-Kac integral representation of the density matrices of the Gibbs state in terms of integrals over trajectories of Wiener processes labelled by sites of the frustrated lattice. Various probabilistic constructions will be involved, and the final outcome will be a theorem that the density matrices are invariant relative the action of symmetry group on trajectories. This will imply that the Gibbs state is invariant. In the case of a non-compact symmetry group, it would mean that a Gibbs state with the initial Hamiltonian does not exist. A separate (and promising) direction is to investigate absense of symmetry-breaking in the ground state of the quantum-gravity Hamiltonian [1,3]. (2) Information Theory and Applied Probability. The fundamental monograph “Information and Coding by Example” contains some refining of previously published results (cf. [9]). Other topics of research in Probability Theory: insurance models [5], branching diffusion processes [10], etc. (3) Mathematical Epidemiology. The small initial contagion approximation helps to solve the stochastic problem as well the deterministic one. The idea is to account for the discreetness of population when number of infectives is small and when the equations for probability distribution can be simplified. At the time of developed outbreak in the given SIR centre when the number of all species are large we switch to the deterministic equation. To confirm analytical results, a large number of computer simulation have been conducted. The direct simulation is compared with the approximated models. The numerical simulation confirms that the proposed approximating is working well if the population of the epidemic centre is large enough. To derive probability distribution of number of species at initial stage (when the number of infective is small) a Markov chain is studied and the Master/Kolmogorov equation for the probability generated function are derived. They are solved analytically for some cases [6-8]. Selected Publications [1] Kelbert M., Suhov Y., Yambartsev A., A Mermin-Wagner theorem on Lorenzian triangulations with quantum spins, Brazilian Journ. Probab. (to appear), ArXiv:1211.5446v1

[2] Kelbert M., Suhov Y., On quantum Mermin-Wagner theorem for a generalized Hubbard model, Advances Math. Phys., V.2013, ID 637375, 2013 [3] Kelbert M., Suhov Y., Yambartsev A., A Mermin-Wagner theorem for Gibbs state on Lorenzian triangulations, Journ. Statist. Phys., V. 150, N.4, 2013. 671–677 [4] Kelbert M., Suhov Y., On quantum Mermin-Wagner theorem for quantum rotators on two-dimensional graphs, Journ. of Mathematical Physics, V. 54, N.3, 2013 [5] Kelbert M., Sazonov I., Avram F., Uniform asymptotics of ruin probabilities for Lévy processes, Markov Processes Related Fields, V.18, 2012, N.4, 681–692 [6] Sazonov I., Kelbert M., Gravenor M.B., A two-stage model for the SIR outbreak: accounting for the discrete nature of the epidemic at the contamination stage, Mathematical Bioscience. V.234, N.2, 2011, 108–117 [7] Kelbert M., Sazonov I., Gravenor M.B., Critical reaction time during a disease outbreak, Ecological Complexity, V.8, 2011, N.4, 326–335 [8] Sazonov I., Kelbert M., Gravenor M.B., Travelling waves in a lattice of SIR nodes in approximation of small coupling, Mathematical Medicine and Biology, V.28, 2011, N.2, 165–183 [9] Kelbert M., Suhov Y., Continuity of mutual entropy in the large signal-to-noise ratio limit. Stochastic Analysis 2010, Berlin: Springer, 2010, 281–299 [10] Kelbert M., Suhov Y., Asymptotic behaviour of a branching diffusion on a hyperbolic space, Theory Probab. Appl., SIAM: V.52, 2008, N.4, 594–613 Eugene Lytvynov (1) Determinantal point processes. Determinantal point processes naturally occur in numerous fields of mathematics. The well known Macchi–Soshnikov theorem establishes a necessary and sufficient condition of existence of a determinantal point process when its correlation operator K is self-adjoint. When studying harmonic analysis of both the infinite symmetric group and the infinite-dimensional unitary group, Borodin and Olshanski derived three classes of determinantal point processes whose correlation operator K is J-selfadjoint, i.e., it is self-adjoint with respect to an indefinite scalar product. In particular, such determinantal point processes occur in Borodin and Olshanski’s paper of 2005 in Ann. of Math. In paper [10], EL derived a necessary and sufficient condition of existence of a determinantal point process with a J-self-adjoint correlation operator. (2) Dynamics of binary jumps in continuum. An important example of a Markov dynamics on lattice configurations is the exclusion process. In this process,

particles randomly hop over the lattice under the only restriction to have no more than one particle at each site of the lattice. This process may have a Bernoulli measure as an invariant (and even symmetrizing) measure, but corresponding stochastic dynamics has nontrivial properties and possess an interesting and reach scaling limit behavior. A straightforward generalization of the exclusion process to the continuum is not possible, because the exclusion restriction (yielding an interaction between particles) obviously disappears for configurations in continuum. In papers [6, 7], a dynamics of binary jumps in continuum was constructed, in which at each jump time two points of the (infinite) configuration change their positions in the Euclidean space. Randomness for choosing a pair of points provides a random interaction between particles of the system. In paper [6], the equilibrium case was discussed when a Poisson measure is symmetrizing for the dynamics. In paper [7], a non-equilibrium process was discussed, and a Vlasov-type scaling of this dynamics was derived. (3) Noncommutative Lévy processes. In [3, 4] a wide class of noncommutative generalized stochastic processes with freely independent values was studied. The corresponding family of noncommutative orthogonal polynomials of infinitely many variables has many interesting properties. In many ways, these polynomials (in fact, operators) resemble the families of orthogonal polynomials of infinitely many variables which belong to the Meixner class in classical probability. So it is natural to call the class of these noncommutative polynomials free Meixner. The concept of a generating function of a family of noncommutative orthogonal polynomials was proposed. This function is operator-valued and involves integrals of noncommuting operators. The generating function was derived for the polynomials from the free Meixner class. This function has a resolvent form and resembles the generating function in the classical case. Anyon statistics forms a continuous bridge between boson and fermion statistics. An anyon statistics is described through q-commutation relations where q is a complex number of modulus 1. The case q = 1 corresponds to Canonical Commutation Relations (bosons), and the case q = −1 corresponds to Canonical Anticommutation Relations (fermions). A representation of the q-commutation relations in the Fock space of q-symmetric functions was given by Goldin and Majid in 2004. In paper [8], anyon (noncommutative) Brownian motion and Poisson process were introduced and discussed as families of linear operators in the Fock space of q-symmetric functions. The notion of anyon independence was proposed and corresponding noncommutative Lévy processes were derived. Selected Publications [1] Kondratiev, Y, Lytvynov, E, Röckner, M, ‘Non-equilibrium stochastic dynamics in continuum: The free case’, Condensed Matter Physics, 11, pp. 701–721, (2008)

[2] Kondratiev, YG, Kutoviy, OV, Lytvynov, E, ‘Diffusion approximation for equilibrium Kawasaki dynamics in continuum’, Stochastic Processes and Their Applications, 118, 1278–1299, (2008) [3] Bozejko, M, Lytvynov, E, ‘Meixner class of non-commutative generalized stochastic processes with freely independent values. I. A characterization’, Communication in Mathematical Physics, 292, 99–129, (2009) [4] Bozejko, M, Lytvynov, E, ‘Meixner class of non-commutative generalized stochastic processes with freely independent values. II. The generating function’, Communication in Mathematical Physics, 302, pp. 425–451, (2011) [5] Li, G, Lytvynov, E, A note on equilibrium Glauber and Kawasaki dynamics for permanental point processes, Methods of Functional Analysis and Topology, 17, pp. 29–46, (2011) [6] Finkelshtein, DL, Kondratiev YG, Kutovyi, OV, Lytvynov, E, ‘Binary jumps in continuum. I. Equilibrium processes and their scaling limits’, Journal of Mathematical Physics, 52, 063304, 25 pp, (2011) [7] Finkelshtein, DL, Kondratiev YG, Kutovyi, OV, Lytvynov, E, ‘Binary jumps in continuum. II. Non-equilibrium process and a Vlasov-type scaling limit’, Journal of Mathematical Physics, 52, 113301, 27 pp (2012) [8] Bozejko, M, Wysoczanski, J, Lytvynov, E, ‘Non-commutative Lévy processes for generalized (particularly anyon) statistics’, Communication in Mathematical Physics, 313, pp. 535–569, (2012) [9] Lytvynov, E, Olshanski, G, ‘Equilibrium Kawasaki dynamics and determinatal point processes’, Journal of Mathematical Sciences, 190, pp. 451–458, (2013) [10] Lytvynov, E, ‘Determinantal point processes with J-Hermitian correlation kernels’, Annals of Probability, 41, pp. 2513–2543, (2013)

Andrew Neate My research interests focus on probabilistic methods in mathematical physics particularly the relation between classical and quantum mechanics. Past topics include: (1) The semiclassical Coulomb/Kepler problem. (Joint work with A. Truman and R. Durran) This work began [4] by considering the “atomic elliptic state”; a particular a stationary quantum state for the Coulomb problem that is concentrated on an ellipse. By considering the associated stochastic mechanics we have been able to provide the first derivation of classical Keplerian motion on the ellipse from a stationary state in the semiclassical limit, a problem which dates all the way back to Schrodinger. This has lead to the study of a “Keplerian diffusion”. In a subsequent paper [5] we have proven (with F.Y. Wang) that the generator of the Keplerian diffusion has a spectral gap.

This work has lead to several papers investigating various aspects of semiclassical Coulomb problems and applications to the formation of planets [1]. Current work in this area focuses on extending the semiclassical mechanics to the parabolic and hyperbolic cases. (2) Semiclassical asymptotics for heat and Burgers equations. (Joint work with A. Truman) This work has its origin in the Hamilton-Jacobi theory developed by A. Truman and D. Elworthy to investigate the semiclassical limit of heat and Schrodinger equations using classical mechanics. These ideas have been developed to include stochastic heat and Burgers equations and I initially worked on the analysis of the behaviour of singularities in the solution for the stochastic Burgers equation [6]. More recently we have worked on extending these results to heat and Burgers equations including vector potentials as well as systems driven by random potentials [2,3]. Current research interests cover the above areas but also include stochastic analysis and graded algebras (joint with E. Beggs) and phase space path integrals (joint with A. Truman). Selected Publications [1] Andrew Neate & Aubrey Truman (2013). A stochastic Burgers-Zeldovich model for the formation of planetary ring systems and the satellites of Jupiter and Saturn. Journal of Mathematical Physics, 54, 033512. doi:10.1063/1.4794514 [2] Andrew Neate & Aubrey Truman (2012). Hamilton-Jacobi theory and the stochastic elementary formula. New trends in stochastic analysis and related topics. A volume in honor of Professor K. D. Elworthy. [3] Andrew Neate, Scott Reasons & Aubrey Truman (2011). The stochastic Burgers equation with vorticity: Semiclassical asymptotic series solutions with applications. Journal of Mathematical Physics, 52, 083512-1. doi:10.1063/1.3610668 [4] Richard Durran, Andrew Neate & Aubrey Truman (2008). The divine clockwork: Bohr’s correspondence principle and Nelson’s stochastic mechanics for the atomic elliptic state. Journal of Mathematical Physics , 49 , 032102. doi:10.1063/1.2837434 [5] Richard Durran, Andrew Neate, Aubrey Truman & Feng-Yu Wang (2008). On the divine clockwork: The spectral gap for the correspondence limit of the Nelson diffusion generator for the atomic elliptic state. Journal of Mathematical Physics , 49 , 102103. doi:10.1063/1.2988715 [6] Andrew Neate & Aubrey Truman (2007). A one-dimensional analysis of turbulence and its intermittence for the d-dimensional stochastic Burgers equation. Markov Process and Related Fields , 13 - 238.

Irina Rodionova IR studied noncommutative analogs of the Meixner class of orthogonal polynomials on the real line. Recall that orthogonal polynomials from the classical Meixner class are characterized by their generating function which has exponential form. This class contains Hermite polynomials (orthogonal with respect to a Gaussian distribution), Charlier polynomials (orthogonal with respect to a Poisson distribution), Laguerre polynomials (orthogonal with respect to a gamma distribution), and two kinds of Meixner polynomials (orthogonal with respect to a negative binomial distribution and a Meixner distribution, respectively). In paper [2], an analog of these polynomials was studied which arise in free probability. In the latter case, the generating function of the orthogonal polynomials on the real line has a generating function of exponential form. In paper [3], a system of orthogonal polynomials of infinitely many noncommuting variables was discussed, which corresponds to an anyon statistics. These polynomials were obtained through an orthogonalization of polynomials of an anyon Lévy process. In fact, the orthogonalization procedure gives to a unitary isomorphism between the noncommutative $L^2$ space of an anyon Lévy process and an extended anyon Fock space. The action of the field operators in this extended anyon Fock space naturally leads to a class of orthogonal polynomials which are of Meixner type, i.e., which are, in a sense, similar to classical orthogonal polynomials of infinitely many variables which have exponential generating function. Publications [1] I. Rodionova, Analysis connected with generating functions of exponential type in one and infinite dimensions, Methods Funct. Anal. Topology 11 (2005), 275–297 . [2] E. Lytvynov, I. Rodionova, Lowering and raising operators for the free Meixner class of orthogonal polynomials, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 387–399. [3] M. Bozejko, E. Lytvynov, I. Rodionova, An extended anyon Fock space and noncommutative Meixner-type orthogonal polynomials, arXiv preprint, 2013.

Feng-Yu Wang (1) Functional inequalities and applications. FYW developed a general theory of functional inequalities, Markov semigroups and spectral theory. The general theory has been applied to specific models to derive new results on essential spectrum, contractivity properties and compactness of semigroups, and long

/short time behaviors of Markov processes. See the monograph of FYW “Functional Inequalities, Markov Semigroups and Spectral Theory” published by Science Press, 2005 (2) Applications of stochastic approaches to geometry analysis. FYW has developed the coupling method to estimate the first eigenvalue on Riemannian manifolds, established a new dimension‐free Harnack inequality for diffusion semigroups which has been widely applied in the study of infinite‐dimensional analysis and SDEs, and found a formula for the second fundamental form for the boundary of a Riemannian manifolds by using short time behavior of the Neumann semigroup. This formula leads to a number of equivalent assertions on the Neumann semigroup for the curvature condition and the lower bound of the second fundamental form. Stochastic analysis on the path space over Riemannian manifolds with boundary is also investigated. See the monograph of FYW “Analysis for Diffusion Processes on Riemannian Manifolds” published by World Scientific, 2013. (3) Stochastic partial differential equations. FYW and collaborators derived the existence and uniqueness of a large class of generalized stochastic porous media equations, and established Harnack inequality and applications for the associated semigroup. Gradient estimates, Harnack inequality and applications are also investigated for semi‐linear SDEs with singular coefficients. Together with his collaborations, FYW developed new coupling methods to study derivative formulas and dimension-free Harnack inequalities for various stochastic equations, including SDEs with multiplicative Gaussian noise, non-linear SPDEs, semi-linear SPDEs, SDEs driven by Lévy noise, and functional SDEs/SPDEs. This type of dimension-free Harnack inequality was first found by FYW in 1997 (Probability Theory and Related Fields), and was called Wang’s Harnack inequality in references. Progresses made in this direction has been included in the monograph of FYW “Harnack Inequalities for Stochastic Partial Differential Equations”, Springer, 2013. Selected Publications [1] F.-Y. Wang, Log-Sobolev inequalities: different roles of Ric and Hess, Annals of Probability. 37(2009), 1587–1604. [2] F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on non-convex manifolds, Annals of Probability. 39(2011), 1449–1467. [3] F.-Y. Wang, Log-Sobolev inequality on non-convex manifolds, Advances in Mathematics. 222(2009), 1503–1520. [4] M. Röckner, F.-Y. Wang, Non-monotone stochastic generalized porous media equations, J. Differential Equations, 245(2008), 3898–3935.

[5] A. Guillin, F.-Y. Wang, Degenerate Fokker-Planck Equations: Bismut Formula, Gradient Estimate and Harnack Inequality, J. Differential Equations. 253(2012), 20–40. [6] F.-Y. Wang, From super Poincare to weighted log-Sobolev and entropy-cost inequalities, J. Mathematiques Pures Appl. 90(2008), 270–285. [7] F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Mathematiques Pures Appl. 94(2010), 304–321. [8] F.-Y. Wang, X. Zhang, Derivative formula and applications for degenerate diffusion semigroups, J. Mathematiques Pures Appl. 99(2013), 726–740. [9] G. Da Prato, M. Rockner, F.-Y. Wang, Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups, J. Functional Analysis. 257(2009), 992–1017. [10] F.-Y. Wang, Second fundamental form and gradient of Neumann semigroups, J. Functional Analysi. 256(2009), 3461–3469.

Jiang-Lun Wu Specialities: stochastic partial differential equations and random fields, stochastic analysis over infinite dimensional spaces, Lévy‐type jump processes and their generators, non‐standard measure theory and stochastic analysis, probability theory on algebraic structures, constructive quantum field theory and statistical mechanics, financial mathematics. JLW's research interests cover a wide range of topics in analysis and probability, including stochastic analysis, nonstandard analysis, analysis over infinite dimensional spaces. (1) Nonstandard analysis methods to stochastic calculus, random fields and mathematical physics. JLW's early research had been motivated by problems of mathematical physics, specifically constructive quantum fields and statistical mechanics, with a feature of linking problems of existence of probability measures on infinite dimensional spaces with functional analytic questions to probability theory. He has used nonstandard analysis approach to derive new and sharp results in stochastic calculus and stochastic differential equations in infinite dimensional spaces, hyperfinite flat integrals for Euclidean random fields and large deviation, all rooted in stochastic quantization (cf. the most recent paper [8]). (2) Euclidean and stochastic approaches to constructive quantum field theory. JLW and collaborators constructed new quantum field models with indefinite metric and nontrivial S‐matrix. This is achieved in a tour de force of analysis and probability by replacing Gaussian processes for Nelson's free field with Lévy processes. The resulting theory has a nontrivial S‐matrix and satisfies the Gording‐Wightman axioms. Moreover, Truman and JLW have initiated and

studied a stochastic model involving Lévy flights for scalar conservation laws (cf. [10]). (3) Loeb’s rich measure techniques for stochastic analysis. JLW developed a hyper‐finite dimensional method to stochastic calculus and stochastic differential equations in infinite dimensional spaces, established hyper‐finite flat integrals for Euclidean random fields and large deviations in stochastic quantization. Furthermore, JLW and collaborators developed a Loeb measure framework for joint measurability and duality, which provides a positive framework for Doob's joint measurability conjecture. This paves an interesting rich‐measure framework for mathematical investigations on equilibrium economics and finance in large scale (cf. [9]). (4) Stochastic differential equations and stochastic partial differential equations with Lévy noise. JLW and collaborators initiated to study parabolic stochastic partial differential equations driven by Lévy white noise which led an active research area and stimulated further investigations on noise perturbation of integro‐differential conservation laws, and stochastic nonlinear pseudo‐differential equations and the associated critical nonlinearity exponents (cf. [2,6]). Moreover, JLW and co-authors have investigated the existence of stationary solutions of stochastic differential equations driven by Lévy processes (cf. [5]). (5) Lévy type processes and quantum theory. JLW and collaborators derived (infinitesimal) invariant measures for the Lévy‐type generators; presented the notion of partly divisible probability measures on locally compact Abelian groups (which links to free random variables in non commutative probability theory); he and coauthors established quantum Poisson stochastic calculus; and most recently Feng-Yu Wang and JLW have obtained a new sharp condition for compactness of Schrödinger semigroups linked with relativistic alpha‐stable processes which has been noticed and used by world leading experts in the field (cf. [4]). (6) Stochastic differential equations and nonlinear partial differential equations with applications in financial modelling. JLW and his PhD students developed stochastic optimal control for stochastic differential equations driven by Lévy‐type processes; he and collaborators have characterized the path independence property of Girsanov transformation density for stochastic differential equations which induces a new link of stochastic differential equations and nonlinear parabolic equations. The latter work reveals certain features of equilibrium states in systems of stochastic dynamics, in particular, in those pricing dynamical systems with stochastic volatility (cf. [1,3]). Furthermore, JLW and co-authors established analytic functional inequalities for the solutions of stochastic Burgers equations (cf. [4]). Selected Publications [1] J.-L. Wu and W. Yang, (i) Pricing CDO tranches in an intensity-based model with the mean-reversion approach, Mathematical and Computer Modelling, 52

(2010), 814–825; (ii) Valuation of synthetic CDOs with affine jump-diffusion processes involving Lévy stable distributions, Mathematical and Computer Modelling, 57 (2013), No 3-4: 570–583. [2] J.-L. Wu and B. Xie, On a Burgers type nonlinear equation perturbed by a pure jump Lévy noise, Bulletin des Sciences Math\'ematiques, 136 (2012), 484–506. [3] A. Truman, F.-Y. Wang, J.-L. Wu and W. Yang, A link of stochastic differential equations to nonlinear parabolic equations, SCIENCE CHINA Mathematics, 55 (2012), No 10: 1971–1976. [4] F.-Y. Wang, J.-L. Wu and L. Xu, Log-Harnack inequality for stochastic Burgers equations and applications, Journal of Mathematical Analysis and Applications, 384 (2011), 151–159. [5] S. Albeverio, Z. Brzezniak and J.-L. Wu, Existence of global solutions and invariant measures for stochastic differential equations driven by Poisson type noise with non-Lipschitz coefficients, Journal of Mathematical Analysis and Applications, 371 (2010), 309–322. [6] N. Jacob, A.K. Potrykus and J.‐L. Wu, (i) Solving a non‐linear pseudo-differential equation of Burgers type, Stochastics and Dynamics, 8 (2008), 613‐‐624; (ii) Solving a non-linear stochastic pseudo-differential equation of Burgers type, Stochastic Processes and their Applications,120 (2010), 2447–2467. [7] F.‐Y. Wang and J.‐L. Wu, Compactness of Schrödinger semigroups with unbounded below potentials, Bulletin des Sciences Mathematiques, 132(2008), 679–689. [8] J.‐L. Wu, A hyperfinite flat integral for generalized random fields, Journal of Mathematical Analysis and Applications, 330 (2007), 133–143. [9] S. Albeverio, Y.N. Sun and J.‐L. Wu, Martingale property for empirical processes. Transactions of American Mathematical Society, 359 (2007), 517–527. [10] A. Truman and J.‐L. Wu, On a stochastic nonlinear equation arising from 1‐D integro‐differential scalar conservation laws. Journal of Functional Analysis, 238 (2006), 612–635.

Chenggui Yuan Research Interests: Stochastic differential equations (SDEs), stochastic functional differential equations (SFDEs), functional inequalities, numerical solutions, population dynamics, stochastic hybrid systems and stochastic control.

(1) Stochastic differential equations with Markovian switching. A research monograph written jointly with X Mao and published by Imperial College Press 2006. This book provides the first systematic presentation of the theory of SDEs with Markovian switching. It presents the basic principles at an introductory level but emphasizes current advanced level research trends. The material takes into account all the features of Ito equations, Markovian switching, interval systems and time-lag. The theory developed is applicable in different and complicated situations in many branches of science and industry. (2) Stability in distribution. In many practical situations, we want to know whether or not the probability distribution of a solutions of SDEs will converge weakly to some distribution. Such convergence is called the stability in distribution and the limit distribution is known as a stationary distribution. Since stability in distribution implies the existence and uniqueness of an invariant measure, it is an important topic is for a SDEs. For finite dimensional case, Basak et al. considered such stability for singular diffusion and semi-linear SDEs under regime switching. Recently, together with X. Mao, CY extended by Lyapunov function approaches the results to cover a class of much more general SDEs under regime switching. Since a mild solutions of stochastic partial differential equations (SPDEs) do not have stochastic differential, a significant consequence of this fact is that we can not employ the Ito formula for mild solutions directly in most of our arguments. Therefore, the method used in finite dimensional case is not applicable to investigate stability in distribution of mild solutions to SPDEs and stochastic functional partial differential equations (SFPDEs). By introducing from Ichikawa an approximating system and constructing appropriate metric between transition probability functions of mild solutions, together with J. Bao and A. Truman, CY gave sufficient conditions for stability in distribution of mild solutions of SPDEs and SFPDEs. Moreover, with his co-authors, CY also studied the stability in distribution for numerical solutions.

(3) Numerical solutions of SDEs. Numerical solutions of SDEs is a hot topic in stochastic analysis. With X. Mao, CY developed the numerical scheme for a class of hybrid systems, which is a SDEs with Markovian switching. In the paper, strong convergence of Euler scheme has been proved for SDEs with Markovian switching under a local Lipschitz condition. Written jointly with DJ Higham and X Mao and published in Siam Journal on Numerical Analysis, CY proved that the Euler Maruyama method correctly reproduces almost sure and moment exponential stability on scalar linear SDEs. With N Jacob and Y Wang, CY obtained the rate of strong convergence of numerical scheme for SDEs with jumps under a local Lipschitz conditions. Recently, with J. Bao, CY investigated the long-term behavior of certain numerical scheme for a class of SPDEs.

(4) The Harnack inequalities for SFDEs with multiplicative noise and applications. The dimension free Harnack inequality has become a useful tool in the study of diffusion semigroups, in particular, for the uniform integrability, contractivity properties, and estimates on heat kernels. Recently, by using coupling arguments, the dimension-free Harnack inequality has been established for SDEs with multiplicative noise, and for SDDEs with additive noise. With F.Y.

Wang, CY extended these existed results to the functional solution of SDDEs with multiplicative noise. Due to the double difficulty caused by delay and non-constant diffusion coefficient, both couplings constructed in the existed literature are no longer valid. Under a reasonable assumption, we constructed a successful coupling which leads to the dimension free Harnack inequality an explicit log-Harnack inequality of the functional solution. This version of Harnack inequality is powerful enough to imply some regularity properties of the semigroup such as the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures. Selected Publications [1] J. Bao and C.Yuan, Convergence rate of EM scheme for SDDEs, Proc. Amer. Math. Soc. , 141 (2013), 3231–3243. [2] J. Bao, F. Y. Wang, C.Yuan, Bismut formulae and applications for functional SPDEs, Journal of Bulletin des Sciences Mathematiques 137(2013), 509–522. [3] C.Yuan and J. Bao, On the Exponential stability for switching-diffusion processes with jumps, Quarterly of Applied Mathematics 71 (2013), 311–329. [4] J. Bao, F. Y. Wang, C.Yuan, Derivative formula and Harnack inequalityfor degenerate functional SDEs, Stochastics & Dynamics 13 (2013), pp 22. [5] J. Shao, F.Y. Wang, C.Yuan, Harnack inequalities for stochastic (functional) differential equations with non-Lipschitzian coefficients.Electron. J. Probab. 17(2012), pp18. [6] J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Applicandae Mathematicae, 116 (2011), 119–132 [7] F.Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stochastic Processes and Applications, 121 (2011), 2692–2710. [8] J. Bao, A. Truman and C. Yuan, Almost sure asymptotic stability of stochastic partial differential equations with jumps, SIAM J. Control Optim. 49, 771–787 (2011). [9] G. Yin, X. Mao, C. Yuan and D. Cao, Approximation methods for hybrid diffusion systems with state-dependent switching processes: numerical algorithms and existence and uniqueness of solutions, SIAM J. Math. Anal. 41(2010) 2335–2352. [10] Mao, X and Yuan, C, Stochastic Differential Equations with Markovian Switching, 409 pages, London , Imperial College Press, 2006.