Infinite Dimensional Analysis, Quantum Probability and Related Topics

In this paper we prove the logarithmic Sobolev inequality on pinned loop groups over compact Lie groups equipped with -metric (s>1/2). We also compute the Ricci tensor with respect to -metric and investigate its asymptotic behavior as s decreases to 1/2.

We introduce and study a notion of Λ-freeness, which generalises both freeness and independence in the context of noncommutative probability. In particular, we extend Voiculescu's construction of the free product of representations of unital *-algebras. A central limit theorem is also proved.

Starting with an array (μ_n,ℓ)_n,ℓ∈ℕ of probability measures on a Lie group G, one forms the convolution products μ_n(s,t):=μ_n,kn(s)+1*⋯*μ_n,kn(t), 0≤s≤t, where k_n:ℝ₊→ℤ₊ are increasing scaling functions. Sufficient conditions are established for convergence μ_n(s,t)→μ(s,t) as n→∞, where (μ(s,t))_0≤s≤t is necessariliy a convolution hemigroup, i.e. μ(s,t)=μ(s,r)*μ(r,t), 0≤s≤r≤t. Using previous results concerning convolution hemigroups of finite variation (Heyer and Pap⁹), this leads to a bijection between the set of convolution hemigroups and an appropriate parameter set containing time-dependent Lévy–Khinchin type triplets. Hence, at the same time, we parametrize the set of measures generated by stochastically continuous processes with independent left increments in G. The above-mentioned sufficient conditions turn out to be necessary and sufficient for convergence of a sequence of random step functions corresponding to the triangular array towards a stochastically continuous process with independent left increments. Connections with results of Feinsilver⁴ and Kunita¹¹ are discussed.

We consider a Markov process X in a Hilbert space H, solution of a semilinear stochastic evolution equation driven by an infinite-dimensional Wiener process, occurring in the equation as an additive noise. Using techniques of the Malliavin calculus, under suitable assumptions, we prove an integration by parts formula for the transition probabilities ν_t, t>0 (the laws of X_t). We deduce results on differentiability (i.e. existence of logarithmic derivatives) of ν_t along a set of directions h∈H which can be described in terms of the coefficients of the equation. The general results are then applied to various classes of non linear stochastic partial differential equations and systems.

A general method for deriving Girsanov or quasi-invariance formulas for classical stochastic processes with independent increments obtained as components of Lévy processes on real Lie algebras is presented. Letting a unitary operator arising from the associated factorizable current representation act on an appropriate commutative subalgebra, a second commutative subalgebra is obtained. Under certain conditions the two commutative subalgebras lead to two classical processes such that the law of the second process is absolutely continuous w.r.t. to the first. Examples include the Girsanov formula for Brownian motion as well as quasi-invariance formulas for the Poisson process, the Gamma process,^15,16 and the Meixner process.

Kuo's Fourier–Mehler transforms {ℱ_Θ} form a one-parameter automorphism group of the space of white noise distributions . The adjoint transforms form a one-parameter automorphism group of the space of test white noise functions but do not admit unitary extensions with respect to the L²-norm induced from the Gaussian measure with variance 1. We prove that admits an extension to a one-parameter group of unitary operators on the L²-space with respect to the Gaussian measure with variance 1/2.