Infinite Dimensional Analysis, Quantum Probability and Related Topics

We prove a very general sharp inequality of the Hölder–Young-type for functions defined on infinite dimensional Gaussian spaces. We begin by considering a family of commutative products for functions which interpolates between the pointwise and Wick products; this family arises naturally in the context of stochastic differential equations, through Wong–Zakai-type approximation theorems, and plays a key role in some generalizations of the Beckner-type Poincaré inequality. We then obtain a crucial integral representation for that family of products which is employed, together with a generalization of the classic Young inequality due to Lieb, to prove our main theorem. We stress that our main inequality contains as particular cases the Hölder inequality and Nelson’s hyper-contractive estimate, thus providing a unified framework for two fundamental results of the Gaussian analysis.

Given a compact Lie group and a conjugate-invariant Levi process on it, generated by the operator $(L,D(L))$, we construct the Levi process on the path space of $G$, associated with the convolution semigroup $\{{\mu }_t,t\ge 0\}$ of probability measures, where ${\mu }_t$ is the distribution of the Levi process on $G$ generated by $(tL,D(L))$. The constructed process is obtained as the weak limit of piecewise constant paths, which, as well as proving its existence and properties, provides finite-dimensional approximations of Chernoff type to the integrals with respect to its distribution.

The partial isometries of ${ℝ}^N,{ℂ}^N$ form compact semigroups ${\tilde{O}}_N,{\tilde{U}}_N$. We discuss here the liberation question for these semigroups, and for their discrete versions ${\tilde{H}}_N,{\tilde{K}}_N$. Our main results concern the construction of half-liberations ${\tilde{H}}_N^{\times },{\tilde{K}}_N^{\times },{\tilde{O}}_N^{\times },{\tilde{U}}_N^{\times }$ and of liberations ${\tilde{H}}_N^{+},{\tilde{K}}_N^{+},{\tilde{O}}_N^{+},{\tilde{U}}_N^{+}$. We include a detailed algebraic and probabilistic study of all these objects, justifying our “half-liberation” and “liberation” claims.

We prove that the Connes embedding problem is stable under graph products.

We consider the solution to the stochastic heat equation driven by the time-space white noise and study the asymptotic behavior of its spatial quadratic variations with “moving time”, meaning that the time variable is not fixed and its values are allowed to be very big or very small. We investigate the limit distribution of these variations via Malliavin calculus.

We study nonlinear $\sigma$-models defined on a noncommutative torus as a two-dimensional string worldsheet. We consider (i) a two-point space, (ii) a circle, (iii) a noncommutative torus, (iv) a classical group $SU(2,ℂ)$ as examples of space-time. Based on established results, the trivial harmonic unitaries of the noncommutative chiral model known as local minima are shown not to be global minima by comparing them to the symmetric unitaries derived from instanton solutions of the noncommutative Ising model corresponding to a two-point space. In addition, a ${\mathbb{Z}}^2$-action on field maps is introduced to a noncommutative torus, and its action on solutions of various Euler–Lagrange equations is described.

We study the existence and regularity of densities for the solution of a nonlinear heat diffusion with stochastic perturbation of Brownian and fractional Brownian motion type: we use the Malliavin calculus in order to prove that, if the nonlinear term is suitably regular, then the law of the solution has a smooth density with respect to the Lebesgue measure.