Barbara Rüdiger

Invariant Measures and Symmetry Property of Lévy Type Operators

Second order elliptic integro-differential operators (Lévy type operators) are investigated. The notion of regular (infinitesimal) invariant probability measures for such operators is posed. Sufficient conditions for the existence of such regular infinitesimal invariant probability measures are obtained and the symmetrization problem is discussed.

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.

Stochastic Integrals and the Lévy–Ito Decomposition Theorem on Separable Banach Spaces

Abstract A direct definition of stochastic integrals for deterministic Banach valued functions on separable Banach spaces with respect to compensated Poisson random measures is given. This definition yields a direct proof of the Lévy–Ito decomposition of a càdlàg process with stationary, independent increments into a jump and a Brownian component. It turns out that if the Lévy measure ν(dx), associated to the compensated Poisson random measure, satisfies ∫0<|x|≤1|x|ν(dx) < ∞, or ∫0<|x|≤1|x|2ν(dx) < ∞ and (in the second case) the Banach space is of type 2, then the pure jump martingale part in the decomposition is a stochastic integral of the function f(x) = x, in a stronger sense than in the decomposition given by Ito [Ito, K. On stochastic processes I (Infinitely divisible laws of probability). J. Math. 1942, 18, 261–301] resp. Dettweiler [Dettweiler, E. Banach space valued processes with independent increments and stochastic integrals. In Probability in Banach spaces IV, Proc., Oberwolfach 1982, Lectures Notes Maths., Springer: Berlin, 1982; 54–83], for the real resp. Banach valued case.

Dynamical Fluctuations at the Critical Point: Convergence to a Nonlinear Stochastic PDE

We consider an Ising spin system with Glauber dynamics and Kac interactions in one dimension at the critical temperature. We study the fluctuation filed of the magnetization density in a scaling limit which involves space, time and the range of the interaction. We prove that for a suitable choice of the scalings the normalized fluctuations field converges to the solution of a one-dimensional (nonlinear) Ginzburg–Landau equation perturbed by a white noise process.

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.

Infinite-dimensional stochastic differential equations obtained by subordination and related Dirichlet forms

Abstract New results related to the decomposition theorem of additive functionals associated to quasi-regular Dirichlet forms are presented. A characterization of subordinate processes associated to quasi-regular symmetric Dirichlet forms in terms of the unique solutions of the corresponding martingale problems is obtained. The subordinate of (generalized) Ornstein–Uhlenbeck processes are exhibited explicitly in terms of generators, Dirichlet forms, and unique pathwise solutions of stochastic differential equations (SDEs). In the case where the state space is infinite dimensional as, e.g. in Euclidean quantum field theory, the construction provides a characterization of the processes in terms of projections on the topological dual space, and corresponding finite-dimensional SDEs.

Analytic and Probabilistic Aspects of Lévy Processes and Fields in Quantum Theory

A review of work on the description of generators and processes associated with stochastic (partial or pseudo-) differential equations driven by general white noises (including jump as well as diffusion parts) is given. Processes with finite- or infinite-dimensional state space are described in a unified way using the theory of Dirichlet forms, combined with the technique of subordination of processes. In particular the analytic problems arising from subordinating sub-Markov semigroups are described. As examples the subordination of stochastic quantization processes is presented. It is also described how stochastic partial differential or pseudodifferential equations are used to construct relativistic quantum fields in indefinite metric with nontrivial scattering in four space- time dimensions.

Time dependent critical fluctuations of a one dimensional local mean field model

SummaryOne-dimensional stochastic Ising systems with a local mean field interaction (Kac potential) are investigated. It is shown that near the critical temperature of the equilibrium (Gibbs) distribution the time dependent process admits a scaling limit given by a nonlinear stochastic PDE. The initial conditions of this approximation theorem are then verified for equilibrium states when the temperature goes to its critical value in a suitable way. Earlier results of Bertini-Presutti-Rüdiger-Saada are improved, the proof is based on an energy inequality obtained by coupling the Glauber dynamics to its voter type, linear approximation.

Exponential Ergodicity of the Jump-Diffusion CIR Process

In this paper we study the jump-diffusion CIR process (shorted as JCIR), which is an extension of the classical CIR model. The jumps of the JCIR are introduced with the help of a pure-jump Levy process \((J_t, t \ge 0)\). Under some suitable conditions on the Levy measure of \((J_t, t \ge 0)\), we derive a lower bound for the transition densities of the JCIR process. We also find some sufficient conditions under which the function \(V(x)=x\), \(x\ge 0\), is a Forster-Lyapunov function for the JCIR process. This allows us to prove that the JCIR process is exponentially ergodic.

Subordination of symmetric quasi -regular Dirichlet forms

The generators of subordinate symmetric (sub-) Markov processes and their domains are exhibited by using spectral theory. The construction preserves sets of essential self -adjointness of the generators. General non local symmetric quasi regular Dirichlet forms and the corresponding processes (with jumps) are shown to be constructible by subordination of processes properly associated to symmetric quasi regular Dirichlet forms (in particular local ones). It is proven that subordination preserves the property of a process to be a symmetric m -tight special standard process. A characterization of the subordinate processes in terms of solutions of the corresponding martingale problems is obtained.