Motivated by application to quantum physics, anticommuting analogues of Wiener measure and Brownian motion are constructed. The corresponding Itô integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established. This machinery is used to provide a Feynman-Kac formula for a class of Hamiltonians. Several specific examples are cons...

Quasi-invariance under translation is established for the σ-finite measure unifying Brownian penalisations, which has been introduced by Najnudel, Roynette and Yor ([10]). For this purpose, the theory of Wiener integrals for centered Bessel processes, due to Funaki, Hariya and Yor ([5]), plays a key role.

In this paper we derive a formula for the expectation of random Hermite polynomials in Skorohod integrals, extending classical results in the adapted case. As an application we recover, under simple conditions and with short proofs, the anticipative Girsanov identity and quasi-invariance results obtained in [6] for quasi-nilpotent shifts on the Wiener space.

A stochastic calculus similar to Malliavin’s calculus is worked out for Brownian excursions. The analogue of the Malliavin derivative in this calculus is not a differential operator, but its adjoint is (like the Skorohod integral) an extension of the Itô integral. As an application, we obtain an expression for the integrand in the stochastic integral representation of square integrable Wiener f...

An estimate about multiple stochastic integrals with respect to a normalized empirical measure Péter Major Summary: Let a sequence of iid. random variables ξ 1 ,. .. , ξ n be given on a measurable space (X, X) with distribution µ together with a function f (x 1 ,. .. , x k) on the product space (X k , X k). Let µ n denote the empirical measure defined by these random variables and consider the ...

We study the Gaussian Radon transform in the classical Wiener space of Brownian motion. We determine explicit formulas for transforms of Brownian functionals specified by stochastic integrals. A Fock space decomposition is also established for Gaussian measure conditioned to closed affine subspaces in Hilbert spaces.

We construct arbitrary matrix elements of the quantum evolution operator for a wide class of self-adjoint canonical Hamiltonians, including those which are polynomial in the Heisenberg operators, as the limit of well defined path integrals involving Wiener measure on phase space, as the diffusion constant diverges. A related construction achieves a similar result for an arbitrary spin Hamiltonian.

In a recent work [1,2,3] this author showed that the diffraction by an impenetrable wedge having arbitrary aperture angle always reduces to a standard Wiener-Hopf factorization. However, he encountered some difficulties in ascertaining the coincidence of WienerHopf solutions with the ones obtained by the Malyuzhinets method. These difficulties are due to the use of two different spectral repres...

Kac integral [1, 2, 3] appears as a path-wise presentation of Brownian motion and shortly becomes, with Feynman approach [4], a powerful tool to study different processes described by the wave-type or diffusion-type equations. In the basic papers [1, 4], the paths distribution was based on averaging over the Wiener measure. It is worthwhile to mention the Kac comment that the Wiener measure can...

We obtain new bounds for the solution of variance-gamma (VG) Stein equation that are correct form approximations in terms Wasserstein and Kolmogorov metrics. These hold all parameters values four parameter VG class. As an application we explicit distance error a six moment theorem approximation double Wiener-Itô integrals.