关键词: Guichardet-Fock空间, 修正随机梯度, 修正点态随机梯度, 修正点态随机梯度的共轭

Abstract: The paper investigate the properties of the modified stochastic gradient operator and modified point-state stochastic gradient operators {_s;s∈R₊} in continuous-time Guichardet-Fock space L2(Γ;η). We show that the modified stochastic gradient operator is a unbounded, densely defined linear operator in L2(Γ;η); the family of modified point-state stochastic gradient operators {_s;s∈R₊} and its adjoint {^*_s;s∈R₊} are bounded linear operator, which have many properties. For example, they satisfies the canonical anti-commutation relations(CAR)and nilpotency; _s^*_s=^*_ss, for ∠s≠t, which means that, the family of operators{_s;s∈R₊} and {^*_s;s∈R₊} are commutive for ∠s≠t; the operator {^*_ss;s∈R₊} is a family of orthogonal projections on L2(Γ;η). Meanwhile, we construct a unitary operator group on L2(Γ;η) with the point-state modified stochastic gradient {_s;s∈R₊} and its adjoint {^*_s;s∈R₊}.

Key words: Guichardet-Fock space, modified stochastic gradient, modified point state stochastic gradient, the adjoint of modified point state stochastic gradient