Cryptography with chaos

Analysis in Banach Spaces

In this paper we construct a theory of stochastic integration of processes with values in L(H, E), where H is a separable Hilbert space and E is a UMD Banach space (i.e., a space in which martingale differences are unconditional). The integrator is an H-cylindrical Brownian motion. Our approach is based on a two-sided L P -decoupling inequality for UMD spaces due to Garling, which is combined with the theory of stochastic integration of L(H, E)-valued functions introduced recently by two of the authors. We obtain various characterizations of the stochastic integral and prove versions of the Ito isometry, the Burkholder-Davis-Gundy inequalities, and the representation theorem for Brownian martingales.

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Stochastic integration in UMD Banach spaces

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Stochastic evolution equations in UMD Banach spaces

Let H be a separable real Hilbert space and let E be a real Banach space. In this paper we construct a stochastic integral for certain operator-valued functions Φ : (0, T ) → L (H,E) with respect to a cylindrical Wiener process {WH(t)}t∈[0,T ]. The construction of the integral is given by a series expansion in terms of the stochastic integrals for certain E-valued functions. As a substitute for the Itô isometry we show that the square expectation of the integral equals the radonifying norm of an operator which is canonically associated with the integrand. We obtain characterizations for the class of stochastically integrable functions and prove various convergence theorems. The results are applied to the study of linear evolution equations with additive cylindrical noise in general Banach spaces. An example is presented of a linear evolution equation driven by a one-dimensional Brownian motion which has no weak solution. In this paper we construct a theory of stochastic integration of operator-valued functions with respect to a cylindrical Wiener process. The range space of the operators is allowed to be an arbitrary real Banach space E. A stochastic integral of this type can be used for solving the linear stochastic Cauchy problem (0.1) dU(t) = AU(t) dt+B dWH(t), t ∈ [0, T ], U(0) = u0. Here, A is the infinitesimal generator of a strongly continuous semigroup {S(t)}t>0 of bounded linear operators on E, the operator B is a bounded linear operator from a separable real Hilbert space H into E, and {WH(t)}t∈[0,T ] is a cylindrical H-Wiener process. Formally, equation (0.1) is solved by the stochastic convolution process (0.2) U(t) = S(t)u0 + ∫ t 0 S(t− s)B dWH (s). It is well known that the integral on the right hand side can be interpreted as an Itô stochastic integral if E is a Hilbert space. A comprehensive theory of abstract stochastic differential equations in Hilbert spaces is presented in the monograph by Da Prato and Zabczyk [5]. More generally the integral can be defined for spaces E with martingale type 2. This has been worked out by Brzeźniak [2]. Examples of martingale type 2 spaces are Hilbert spaces and the Lebesgue spaces L(μ) with p ∈ [2,∞). In both settings, the integral is defined for step functions first, and for general functions the integral is obtained by a limiting argument. Such a limiting argument depends on 2000 Mathematics Subject Classification. Primary: 60H05; Secondary: 28C20, 35R15, 47D06, 60H15.

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Stochastic integration of functions with values in a Banach space

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On the space of vector-valued functions integrable with respect to the white noise

The work of the Brussels-Austin group on irreversibility over the last years has shown that quantum large Poincare systems with diagonal singularity lead to an extension of quantum theory beyond the conventional Hilbert space framework and logic. We characterize the algebra of observables, the states and the logic of the extended quantum theory of intrinsically irreversible systems with diagonal singularity. We illustrate the general ideas for the Friedrichs model.

Intrinsic irreversibility of quantum systems with diagonal singularity

Deterministic diffusion, De Rham equation and fractal eigenvectors

Ergodic properties of piecewise linear maps on fractal repellers

We consider the problem of rigging for the Koopman operators of the Renyi and the baker maps. We show that the rigged Hilbert space for the Renyi maps has some of the properties of a strict inductive limit and give a detailed description of the rigged Hilbert space for the baker maps.

Z. Suchanecki

Papers

On the space of vector-valued functions integrable with respect to the white noise

Intrinsic irreversibility of quantum systems with diagonal singularity

Deterministic diffusion, De Rham equation and fractal eigenvectors

Ergodic properties of piecewise linear maps on fractal repellers

Rigged Hilbert spaces for chaotic dynamical systems