q-Gaussian processes are deformations of the usual Gaussian distribution. There are several different versions of this; here we treat a multivariate deformation, also addressed as q-Gaussian process, arising from free probability theory and corresponding to deformations of the canonical commutation relations. For other deformations of Gaussian distributions, see q-Gaussian distribution and Gaussian q-distribution.
History The q-Gaussian process was formally introduced in a paper by Frisch and Bourret under the name of parastochastics, and also later by Greenberg as an example of infinite statistics. It was mathematically established and investigated in papers by Bozejko and Speicher and by Bozejko, Kümmerer, and Speicher in the context of non-commutative probability. It is given as the distribution of sums of creation and annihilation operators in a q-deformed Fock space. The calculation of moments of those operators is given by a q-deformed version of a Wick formula or Isserlis formula. The specification of a special covariance in the underlying Hilbert space leads to the q-Brownian motion, a special non-commutative version of classical Brownian motion.
q-Fock space In the following q ∈ [ − 1 , 1 ] {\displaystyle q\in [-1,1]} is fixed. Consider a Hilbert space H {\displaystyle {\mathcal {H}}} . On the algebraic full Fock space
F alg ( H ) = ⨁ n ≥ 0 H ⊗ n , {\displaystyle {\mathcal {F}}_{\text{alg}}({\mathcal {H}})=\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n},}
where H 0 = C Ω {\displaystyle {\mathcal {H}}^{0}=\mathbb {C} \Omega } with a norm one vector Ω {\displaystyle \Omega } , called vacuum, we define a q-deformed inner product as follows:
⟨ h 1 ⊗ ⋯ ⊗ h n , g 1 ⊗ ⋯ ⊗ g m ⟩ q = δ n m ∑ σ ∈ S n ∏ r = 1 n ⟨ h r , g σ ( r ) ⟩ q i ( σ ) , {\displaystyle \langle h_{1}\otimes \cdots \otimes h_{n},g_{1}\otimes \cdots \otimes g_{m}\rangle _{q}=\delta _{nm}\sum _{\sigma \in S_{n}}\prod _{r=1}^{n}\langle h_{r},g_{\sigma (r)}\rangle q^{i(\sigma )},}
where i ( σ ) = # { ( k , ℓ ) ∣ 1 ≤ k < ℓ ≤ n ; σ ( k ) > σ ( ℓ ) } {\displaystyle i(\sigma )=\#\{(k,\ell )\mid 1\leq k<\ell \leq n;\sigma (k)>\sigma (\ell )\}} is the number of inversions of σ ∈ S n {\displaystyle \sigma \in S_{n}} . The q-Fock space is then defined as the completion of the algebraic full Fock space with respect to this inner product
F q ( H ) = ⨁ n ≥ 0 H ⊗ n ¯ ⟨ ⋅ , ⋅ ⟩ q . {\displaystyle {\mathcal {F}}_{q}({\mathcal {H}})={\overline {\bigoplus _{n\geq 0}{\mathcal {H}}^{\otimes n}}}^{\langle \cdot ,\cdot \rangle _{q}}.}
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