On analytic Ornstein-Uhlenbeck semigroups in infinite dimensions

We extend to infinite dimensions an explicit formula of Chill, Fašangová, Metafune, and Pallara for the optimal angle of analyticity of analytic Ornstein-Uhlenbeck semigroups. The main ingredient is an abstract representation of the Ornstein-Uhlenbeck operator in divergence form. The authors are supported by the ‘VIDI subsidie’ 639.032.201 of the Netherlands Organization for Scientific Research (NWO) and by the Research Training Network HPRN-CT-2002-00281. Received: 28 June 2006 Revised: 5 January 2007     

Approximating a Feller Semigroup by Using the Yosida Approximation of the Symbol of its Generator

We show that if the pseudodifferential operator −q(x,D) generates a Feller semigroup (T_t)_t≥0 then the Feller semigroups (T_t^(v))_t≥0 generated by the pseudodifferential operators with symbol will converge strongly to (T_t)_t≥0 as ν →∞.     

ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS

It is well known that the analysis of the large-time asymptotics of Fokker-Planck type equations by the entropy method is closely related to proving the validity of convex Sobolev inequalities. Here we highlight this connection from an applied PDE point of view.

In our unified presentation of the theory we present new results to the following topics: an elementary derivation of Bakry-Emery type conditions, results concerning perturbations of invariant measures with general admissible entropies, sharpness of convex Sobolev inequalities, applications to non-symmetric linear and certain non-linear Fokker-Planck type equations (Desai-Zwanzig model, drift-diffusion-Poisson model).     

An Approximation for the Zakai Equation

In this article we consider a polygonal approximation to the unnormalized conditional measure of a filtering problem, which is the solution of the Zakai stochastic differential equation on measure space. An estimate of the convergence rate based on a distance which is equivalent to the weak convergence topology is derived. We also study the density of the unnormalized conditional measure, which is the solution of the Zakai stochastic partial differential equation. An estimate of the convergence rate is also given in this case. 60F25, 60H10.} Accepted 23 April 2001. Online publication 14 August 2001.     

Persistence of a two-dimensional super-Brownian motion in a catalytic medium

Summary. The super-Brownian motion X ^ϱ in a super-Brownian medium ϱ constructed in [DF97a] is known to be persistent (no loss of expected mass in the longtime behaviour) in dimensions one ([DF97a]) and three ([DF97b]). Here we fill the gap in showing that persistence holds also in the critical dimension two. The key to this result is that in any dimension (d≤3), given the catalyst, the variance of the process is finite `uniformly in time'. This is in contrast to the `classical' super-Brownian motion where this holds only in high dimensions (d≥3), whereas in low dimensions the variances grow without bound, and the process clusters leading to local extinction. Received: 21 November 1996 / In revised form: 31 March 1997     

Commutator bounds for eigenvalues,with applications to spectralgeometry

We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace– Beltrami operators on manifolds. We significantly improve earlier bounds of Yang and Yau, Li, and Harrell.     

On function spaces related to fractional diffusions on d-sets

We prove that all the Dirichlet forms associated with certain diffusions on a d-set are equivalent and that their common domain is an integral Lipschitz space. We also provide an analytic characterisation of the walk dimension d_w of a d-set F and show that all fractional diffusions on F share d_w as their walk dimension.     

Optimal Control for the Degenerate Elliptic Logistic Equation

We consider the optimal control of harvesting the diffusive degenerate elliptic logistic equation. Under certain assumptions, we prove the existence and uniqueness of an optimal control. Moreover, the optimality system and a characterization of the optimal control are also derived. The sub-supersolution method, the singular eigenvalue problem and differentiability with respect to the positive cone are the techniques used to obtain our results.     

Regularity properties of transition probabilities in infinite dimensions

In a Hilbert space H we consider a process X solution of a semilinear stochastic differential equation, driven by a Wiener process. We prove that, under appropriate conditions, the transition probabilities of X are absolutely continuous with respect to a properly chosen gaussian measure μ in H, and the corresponding densities belong to some Wiener-Sobolev spaces over (H,μ). In the linear caseX is a nonsymmetric Ornstein-Uhlenbeck process, with possibly degenerate diffusion coefficient. The general case is treated by the Girsanov. Theorem and the Malliavin calculus. Examples and applications to stochastic partial differential equations are given     

Filtering of a Markov Jump Process with Counting Observations

This paper concerns the filtering of an R ^d -valued Markov pure jump process when only the total number of jumps are observed. Strong and weak uniqueness for the solutions of the filtering equations are discussed. Accepted 12 November 1999     

Stochastic Quantization of the Two-Dimensional Polymer Measure

We prove that there exists a diffusion process whose invariant measure is the two-dimensional polymer measure ν _g . The diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. We prove the closability of the appropriate pre-Dirichlet form which is of gradient type, using a general closability result by two of the authors. This result does not require an integration by parts formula (which does not hold for the two-dimensional polymer measure ν _g ) but requires the quasi-invariance of ν _g along a basis of vectors in the classical Cameron—Martin space such that the Radon—Nikodym derivatives (have versions which) form a continuous process. We also show the Dirichlet form to be irreducible or equivalently that the diffusion process is ergodic under time translations. Accepted 16 April 1998     

On the mixing property and the ergodic principle for nonhomogeneous Markov chains

We study different types of limit behavior of infinite dimension discrete time nonhomogeneous Markov chains. We show that the geometric structure of the set of those Markov chains which have asymptotically stationary density depends on the considered topologies. We generalize and correct some results from Ganikhodjaev et al. (2006) [3].     

Invariant Measure for Diffusions with Jumps

Our purpose is to study an ergodic linear equation associated to diffusion processes with jumps in the whole space. This integro-differential equation plays a fundamental role in ergodic control problems of second order Markov processes. The key result is to prove the existence and uniqueness of an invariant density function for a jump diffusion, whose lower order coefficients are only Borel measurable. Based on this invariant probability, existence and uniqueness (up to an additive constant) of solutions to the ergodic linear equation are established. Accepted 24 February 1998     

Almost Periodically Distributed Solutions for Diffusion Equations in Duals of Nuclear Spaces

We discuss the problem of the existence of almost periodic in distribution solutions of nuclear space-valued diffusion equations with almost periodic coefficients. Under a dissipativity condition we prove that the translation of the unique mean square bounded solution is almost periodically distributed. Similar results hold in the affine case under mean square stability of the linear part of the equation if the nuclear space is a component of a special compatible family. Accepted 19 December 1996     

Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Large Deviation Principle for Optimal Filtering

   Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

On Interacting Systems of Hilbert-Space-Valued Diffusions

A nonlinear Hilbert-space-valued stochastic differential equation where L ^-1 (L being the generator of the evolution semigroup) is not nuclear is investigated in this paper. Under the assumption of nuclearity of L ^-1 , the existence of a unique solution lying in the Hilbert space H has been shown by Dawson in an early paper. When L ^-1 is not nuclear, a solution in most cases lies not in H but in a larger Hilbert, Banach, or nuclear space. Part of the motivation of this paper is to prove under suitable conditions that a unique strong solution can still be found to lie in the space H itself. Uniqueness of the weak solution is proved without moment assumptions on the initial random variable. A second problem considered is the asymptotic behavior of the sequence of empirical measures determined by the solutions of an interacting system of H -valued diffusions. It is shown that the sequence converges in probability to the unique solution Λ ₀ of the martingale problem posed by the corresponding McKean—Vlasov equation. Accepted 4 April 1996     

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.     

Perturbations of Pseudodifferential Operators with Negative Definite Symbol

Pseudodifferential operators with negative definite symbols appear as generators of jump-type Markov processes. The purpose of this paper is to treat the large jumps of the process by a perturbation approach for the generator. This is of particular interest since in this way the generators are made accessible to a symbolic calculus of pseudodifferential operators. The main auxiliary result consists of a characterization of tightness of the jump measures in terms of the symbol.