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On stochastic differential equations with locally unbounded drift

István Gyöngy, Teresa Martínez (2001)

Czechoslovak Mathematical Journal

We study the regularizing effect of the noise on differential equations with irregular coefficients. We present existence and uniqueness theorems for stochastic differential equations with locally unbounded drift.

Q-adapted quantum stochastic integrals and differentials in Fock scale

Viacheslav Belavkin, Matthew Brown (2011)

Banach Center Publications

In this paper we first introduce the Fock-Guichardet formalism for the quantum stochastic (QS) integration, then the four fundamental processes of the dynamics are introduced in the canonical basis as the operator-valued measures, on a space-time σ-field , of the QS integration. Then rigorous analysis of the QS integrals is carried out, and continuity of the QS derivative D is proved. Finally, Q-adapted dynamics is discussed, including Bosonic (Q = I), Fermionic (Q = -I), and monotone (Q = O) quantum...

Quantum Itô B*-algebras, their classification and decomposition

V. Belavkin (1998)

Banach Center Publications

A simple axiomatic characterization of the general (infinite dimensional, noncommutative) Itô algebra is given and a pseudo-Euclidean fundamental representation for such algebra is described. The notion of Itô B*-algebra, generalizing the C*-algebra, is defined to include the Banach infinite dimensional Itô algebras of quantum Brownian and quantum Lévy motion, and the B*-algebras of vacuum and thermal quantum noise are characterized. It is proved that every Itô algebra is canonically decomposed...

Quantum Lévy-type Laplacian and associated stochastic differential equations

A. Barhoumi, H. Ouerdiane (2006)

Banach Center Publications

We study a quantum extension of the Lévy Laplacian, so-called quantum Lévy-type Laplacian, to the nuclear algebra of operators on spaces of entire functions. We give several examples of the action of the quantum Lévy-type Laplacian on basic operators and we study a quantum white noise convolution differential equation involving the quantum Lévy-type Laplacian.

q-White noise and non-adapted stochastic integral

Un Cig Ji, Byeong Su Min (2006)

Banach Center Publications

The q-white noise is studied as the time derivative of the q-Brownian motion. As an application of the q-white noise, a non-adapted (non-commutative) stochastic integral with respect to the q-Brownian motion is constructed.

Regularity of Gaussian white noise on the d-dimensional torus

Mark C. Veraar (2011)

Banach Center Publications

In this paper we prove that a Gaussian white noise on the d-dimensional torus has paths in the Besov spaces B p , - d / 2 ( d ) with p ∈ [1,∞). This result is shown to be optimal in several ways. We also show that Gaussian white noise on the d-dimensional torus has paths in the Fourier-Besov space b ̂ p , - d / p ( d ) . This is shown to be optimal as well.

Speed of the Brownian loop on a manifold

Rémi Léandre (2006)

Banach Center Publications

We define the speed of the curved Brownian bridge as a white noise distribution operating on stochastic Chen integrals.

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