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- Markov evolutions and hierarchical equations in the continuum. I: one-component systemsPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoGeneral birth-and-death as well as hopping stochastic dynamics of infinite particle systems in the continuum are considered. We derive corresponding evolution equations for correlation functions and generating functionals. General considerations are illustrated in a number of concrete examples of Markov evolutions appearing in applications.
- Glauber dynamics in the continuum via generating functionals evolutionPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoWe construct the time evolution for states of Glauber dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, leading to a local (in time) solution which, under certain initial conditions, might be extended to a global one. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.
- Markov evolutions and hierarchical equations in the continuum. II: multicomponent systemsPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoGeneral birth-and-death as well as hopping stochastic dynamics of infinite multicomponent particle systems in the continuum are considered. We derive the corresponding evolution equations for quasi-observables and correlation functions. We also present sufficient conditions that allow us to consider these equations on suitable Banach spaces.
- An infinite dimensional umbral calculusPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Lytvynov, Eugene; Oliveira, Maria JoãoThe aim of this paper is to develop foundations of umbral calculus on the space $\mathcal D'$ of distributions on $\mathbb R^d$, which leads to a general theory of Sheffer polynomial sequences on $\mathcal D'$. We define a sequence of monic polynomials on $\mathcal D'$, a polynomial sequence of binomial type, and a Sheffer sequence. We present equivalent conditions for a sequence of monic polynomials on $\mathcal D'$ to be of binomial type or a Sheffer sequence, respectively. We also construct a lifting of a sequence of monic polynomials on $\mathbb R$ of binomial type to a polynomial sequence of binomial type on $\mathcal D'$, and a lifting of a Sheffer sequence on $\mathbb R$ to a Sheffer sequence on $\mathcal D'$. Examples of lifted polynomial sequences include the falling and rising factorials on $\mathcal D'$, Abel, Hermite, Charlier, and Laguerre polynomials on $\mathcal D'$. Some of these polynomials have already appeared in different branches of infinite dimensional (stochastic) analysis and played there a fundamental role.
- Kawasaki dynamics in the continuum via generating functionals evolutionPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Oliveira, Maria JoãoWe construct the time evolution of Kawasaki dynamics for a spatial infinite particle system in terms of generating functionals. This is carried out by an Ovsjannikov-type result in a scale of Banach spaces, which leads to a local (in time) solution. An application of this approach to Vlasov-type scaling in terms of generating functionals is considered as well.
- Dynamical widom-rowlinson model and its mesoscopic limitPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Kutoviy, Oleksandr; Oliveira, Maria JoãoWe consider the non-equilibrium dynamics for the Widom–Rowlinson model (without hard-core) in the continuum. The Lebowitz–Penrose-type scaling of the dynamics is studied and the system of the corresponding kinetic equations is derived. In the spacehomogeneous case, the equilibrium points of this system are described. Their structure corresponds to the dynamical phase transition in the model. The bifurcation of the system is shown.
- Results about the free kawasaki dynamics of continuous particle systems in infinite volume: long-time asymptotics and hydrodynamic limitPublication . Kondratiev, Yuri G.; Kuna, Tobias; Oliveira, Maria João; Silva, José Luís da; Streit, LudwigAn infinite particle system of independent jumping particles in infinite volume is considered. Their construction is recalled, further properties are derived, the relation with hierarchical equations, Poissonian analysis, and second quantization are discussed. The hydrodynamic limit for a general initial distribution satisfying a mixing condition is derived. The long-time asymptotics is computed under an extra assumption. The relation with constructions based on infinite volume limits is discussed.
- Analytic aspects of Poissonian white noise analysisPublication . Kondratiev, Yuri G.; Kuna, Tobias; Oliveira, Maria JoãoGeneral structures of Poissonian white noise analysis are presented.Simultaneously, the theory is developed on Poisson and Lebesgue- Poisson space. Both spaces have an own S-transform, well known in the Gaussian case. They give an extra connection between these two spaces via the Bargmann-Segal space. Test and generalized functions,different types of convolutions, and representations of creation and annihilation operators in the aforementioned spaces are considered.
- Sheffer homeomorphisms of spaces of entire functions in infinite dimensional analysisPublication . Finkelshtein, Dmitri L.; Kondratiev, Yuri G.; Lytvynov, Eugene; Oliveira, Maria João; Streit, LudwigFor certain Sheffer sequences $(s_n)_{n=0}^\infty$ on $\mathbb C$, Grabiner (1988) proved that, for each $\alpha\in[0,1]$, the corresponding Sheffer operator $z^n\mapsto s_n(z)$ extends to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$, the Fréchet topological space of entire functions of exponential order $\alpha$ and minimal type. In particular, every function $f\in \mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$ admits a unique decomposition $f(z)=\sum_{n=0}^\infty c_n s_n(z)$, and the series converges in the topology of $\mathcal E^{\alpha}_{\mathrm{min}}(\mathbb C)$. Within the context of a complex nuclear space $\Phi$ and its dual space $\Phi'$, in this work we generalize Grabiner's result to the case of Sheffer operators corresponding to Sheffer sequences on $\Phi'$. In particular, for $\Phi=\Phi'=\mathbb C^n$ with $n\ge2$, we obtain the multivariate extension of Grabiner's theorem. Furthermore, for an Appell sequence on a general co-nuclear space $\Phi'$, we find a sufficient condition for the corresponding Sheffer operator to extend to a linear self-homeomorphism of $\mathcal E^{\alpha}_{\mathrm{min}}(\Phi')$ when $\alpha>1$. The latter result is new even in the one-dimensional case.
- Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spacesPublication . Kondratiev, Yuri G.; Kuna, Tobias; Oliveira, Maria JoãoHarmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L2-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits,in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.