Stochastic integrals: a combinatorial approach

Rota, Gian‐Carlo; Wallstrom, Timothy C.

doi:10.1214/aop/1024404513

Cited by 51 publications

(94 citation statements)

References 28 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…1]). Observe that a random variable of the type [45] for a connection between multiple Wiener-Itô and Hermite polynomials. For every q ≥ 0, we write J q to indicate the orthogonal projection operator on the qth Wiener chaos associated with X , so that, if F ∈ L 2 ( , F , P) is as in (2.27), then J q F = I q ( f q ) for every q ≥ 0.…”

Section: Elements Of Malliavin Calculusmentioning

confidence: 99%

Stein’s method on Wiener chaos

Nourdin

Peccati

2008

Probab. Theory Relat. Fields

351

486

View full text Add to dashboard Cite

Section: Elements Of Malliavin Calculusmentioning

confidence: 99%

Stein’s method on Wiener chaos

Nourdin

Peccati

2008

Probab. Theory Relat. Fields

351

486

View full text Add to dashboard Cite

“…The expansion of Poissonian field moments in terms of cumulants, or more generally the expansion of Green's functions in terms of their connected Green's functions, can best be described in the language of partitions [22].…”

Section: Moments and Cumulantsmentioning

confidence: 99%

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

Gough

2004

Commun. Math. Phys.

View full text Add to dashboard Cite

We consider a Markovian approximation, of weak coupling type, to an open system perturbation involving emission, absorption and scattering by reservoir quanta. The result is the general form for a quantum stochastic flow driven by creation, annihilation and gauge processes. A weak matrix limit is established for the convergence of the interaction-picture unitary to a unitary, adapted quantum stochastic process and of the Heisenberg dynamics to the corresponding quantum stochastic flow: the convergence strategy is similar to the quantum functional central limits introduced by Accardi, Frigerio and Lu [1] . The principal terms in the Dyson series expansions are identified and re-summed after the limit to obtain explicit quantum stochastic differential equations with renormalized coefficients. An extension of the Pulé inequalities [2] allows uniform estimates for the Dyson series expansion for both the unitary operator and the Heisenberg evolution to be obtained.

show abstract

“…The latter have been constructed by Anshelevich [3] following Rota and Wallstrom [11]. Let X be a free infinitely divisible random variable.…”

Section: Free Stochastic Measures and The Hilbert Seriesmentioning

confidence: 99%

A noncrossing basis for noncommutative invariants of SL(2,C)

Lehner

2011

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

Noncommutative invariant theory is a generalization of the classical invariant theory of the action of SL(2, C) on binary forms. The dimensions of the spaces of invariant noncommutative polynomials coincide with the numbers of certain noncrossing partitions. We give an elementary combinatorial explanation of this fact by constructing a noncrossing basis of the homogeneous components. Using the theory of free stochastic measures this provides a combinatorial proof of the Molien-Weyl formula in this setting.Invariant theory has played a major role in 19th century mathematics. It has seen a revival in the last decades and one of the recent generalizations is noncommutative invariant theory. The study of noncommutative invariants of SL(n, C) has been initiated by Almkvist, Dicks, Formanek and Kharchenko [6,5,2], see [1] for a survey. An approach using Young tableaux was realized by Teranishi [15] and the symbolic method was adapted from the classical to the noncommutative setting by Tambour [14]. The latter provides the ground on which we establish a natural basis of the noncommutative invariants which is in bijection with certain noncrossing partitions. It arose after computer experiments and subsequent consulting of Sloane's database [12]. This bijection is applied to provide a combinatorial proof of the Molien-Weyl integral formula for the Hilbert-Poincaré series in this setting, using free cumulants and free stochastic measures.This note is organized as follows. In Section 1 we give a short survey of invariant theory and the statement of the problem. In Section 2 we review a few facts from free probability theory and noncrossing partitions. In Section 3 we explain the symbolic method and construct the noncrossing basis announced in the title. In Section 4 we review the necessary combinatorial aspects of free

show abstract

Stochastic integrals: a combinatorial approach

Cited by 51 publications

References 28 publications

Stein’s method on Wiener chaos

Stein’s method on Wiener chaos

Quantum Flows as Markovian Limit of Emission, Absorption and Scattering Interactions

A noncrossing basis for noncommutative invariants of SL(2,C)

Contact Info

Product

Resources

About