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Contractions and central extensions of quantum white noise Lie algebras

Accardi L., Boukas A.

Probability and Mathematical Statistics

2015

Vol. 35, Fasc. 1

41--72

We show that the Renormalized Powers of Quantum White Noise Lie algebra RPQWN*, with the convolution type renormalization δn(t − s) = δ(s)δ(t − s) of the n ≥ 2 powers of the Dirac delta function, can be obtained through a contraction of the Renormalized Powers of Quantum White Noise Lie algebra RPQWNc with the scalar renormalization δn(t) = cn−1δ(t), c > 0. Using this renormalization, we also obtain a Lie algebra W∞(c) which contains the w∞ Lie algebra of Bakas and the Witt algebra as contractions. Motivated by the W∞ algebra of Pope, Romans and Shen, we show that W∞(c) can also be centrally extended in a non-trivial fashion. In the case of the Witt subalgebra of W∞, the central extension coincides with that of the Virasoro algebra.

Stochastic evolutions driven by non-linear white noise

Accardi L., Boukas A.

Probability and Mathematical Statistics

2002

Vol. 22, Fasc. 1

141--154

We prove the existence and uniqueness theorem for stochastic differential equations with bounded coefficients driven by the renormalized square of white noise.These equations are interpreted as sesquilinear forms on the linear span of the exponential vectors (of the first order white noise) and the existence theorem is establishedon the space of these forms.