The Wiener disorder problem with finite horizon

The Wiener disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the drift of an observed Wiener process changes from one value to another. In this paper we present a solution of the Wiener disorder problem when the horizon is finite. The method of proof is based on reducing the initial problem to a parabolic free-boundary problem where the continuation region is determined by a continuous curved boundary. By means of the change-of-variable formula containing the local time of a diffusion process on curves we show that the optimal boundary can be characterized as a unique solution of the nonlinear integral equation.     

Banach Space-Valued Ornstein–Uhlenbeck Processes with General Drift Coefficients

We construct Ornstein–Uhlenbeck processes with values in Banach space and with continuous paths. The drift coefficient must only generate a strongly continuous semigroup on the Hilbert space which determines the Brownian motion. We admit arbitrary starting points and consider also invariant measures for the process, generalizing earlier work in many directions. A price for the generality is that sometimes one has to enlarge the phase space but most previously known results are covered.The constructions are based on abstract Wiener space methods, more precisely on images of abstract Wiener spaces under suitable linear transformations of the Cameron–Martin space. The image abstract Wiener measures are then given by stochastic extensions. We present the basic spaces and operators and the most important results on image spaces and stochastic extensions in some detail.     

Rough paths analysis of general Banach space-valued Wiener processes

In this article, we carry out a rough paths analysis for Banach space-valued Wiener processes. We show that most of the features of the classical Wiener process pertain to its rough path analog. To be more precise, the enhanced process has the same scaling properties and it satisfies a Fernique type theorem, a support theorem and a large deviation principle in the same Hölder topologies as the classical Wiener process does. Moreover, the canonical rough paths of finite dimensional approximating Wiener processes converge to the enhanced Wiener process. Finally, a new criterion for the existence of the enhanced Wiener process is provided which is based on compact embeddings. This criterion is particularly handy when analyzing Kunita flows by means of rough paths analysis which is the topic of a forthcoming article.     

Gibbs Measures on Mutually Interacting Brownian Paths under Singularities

We are interested in the analysis of Gibbs measures defined on two independent Brownian paths in ?^d interacting through a mutual self‐attraction. This is expressed by the Hamiltonian with two probability measures μ and ν representing the occupation measures of two independent Brownian motions. We will be interested in a class of potentials V that are singular , e.g., Dirac‐ or Coulomb‐type interactions in ?³, or the correlation function of the parabolic Anderson problem with white noise potential. The mutual interaction of the Brownian paths inspires a compactification of the quotient space of orbits of product measures, which is structurally different from the self‐interacting case introduced in [27], owing to the lack of shift‐invariant structure in the mutual interaction. We prove a strong large‐deviation principle for the product measures of two Brownian occupation measures in such a compactification and derive asymptotic path behavior under Gibbs measures on Wiener paths arising from mutually attracting singular interactions. For the spatially smoothened parabolic Anderson model with white noise potential, our analysis allows a direct computation of the annealed Lyapunov exponents, and a strict ordering of them implies the intermittency effect present in the smoothened model. © 2017 Wiley Periodicals, Inc.     

A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.     

Wasserstein space over the Wiener space

The goal of this paper is to study optimal transportation problems and gradient flows of probability measures on the Wiener space, based on and extending fundamental results of Feyel–Üstünel. Carrying out the program of Ambrosio–Gigli–Savaré, we present a complete characterization of the derivative processes for certain class of absolutely continuous curves. We prove existence of the gradient flow curves for the relative entropy w.r.t. the Wiener measure and identify these gradient flow curves with solutions of the Ornstein–Uhlenbeck evolution equation.     

Mosco-convergence and Wiener measures for conductive thin boundaries

The Mosco-convergence of energy functionals and the convergence of associated Wiener measures are proved for a domain with highly conductive thin boundary. We obtain those results for matrix-valued conductivities and a family of speed measures (measures of the underlying domain). In particular, this family includes the Lebesgue measure and the one which makes the energy functional superposition. The expectation of the displacement of the associated processes close to the boundary goes to +∞ due to the explosion of the conductivity at the limit.     

Mixed boundary value problems of diffraction by a half‐plane with an obstacle perpendicular to the boundary

The paper is devoted to the analysis of wave diffraction problems modeled by classes of mixed boundary conditions and the Helmholtz equation, within a half‐plane with a crack. Potential theory together with Fredholm theory, and explicit operator relations, are conveniently implemented to perform the analysis of the problems. In particular, an interplay between Wiener–Hopf plus/minus Hankel operators and Wiener–Hopf operators assumes a relevant preponderance in the final results. As main conclusions, this study reveals conditions for the well‐posedness of the corresponding boundary value problems in certain Sobolev spaces and equivalent reduction to systems of Wiener–Hopf equations. Copyright © 2013 John Wiley & Sons, Ltd.     

Lévy-Gromov's isoperimetric inequality for an infinite dimensional diffusion generator

We establish, by simple semigroup arguments, a Lévy-Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian, isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature.Oblatum 19-VI-1995     

Onsager-Machlup functional for some smooth norms on Wiener space

Summary In this paper, we determine Onsager-Machlup functionals for a variety of norms on Wiener space which includes among others Hölder norms for every 0<<1/2, as well as Besov or Sobolev type norms. We basically require the knowledge of the small ball probabilities for the Wiener measure and use versions of the norms which are rotationaly invariant on the range of the Brownian paths, a property of crucial importance in our approach.     

Energy Minimizing Mountain Ascent

In this article, an optimal mountain ascent is studied as a particular problem of a human walking over a rugged terrain. First, an approximation of the terrain is constructed using particular smooth splines??macro-elements. Then a functional measuring the energy consumption along boundary curves of a macro-element is defined. Finally, the corresponding discrete problem of finding the optimal path on a mesh of curves is applied. Numerical results on real-life data indicate that computed paths are a good approximation of hiking paths in nature.     

The Helmholtz equation in a quadrant with Robin's conditions

The boundary value problem for the Helmholtz equation in a quadrant with different Robin's or impedance‐type conditions is considered by the use of variational methods and boundary pseudodifferential equations. The coerciveness of the underlying sesquilinear form in dependence of the impedance parameters is proved. The equivalence between the boundary value problem and a scalar uniquely solvable Wiener–Hopf–Hankel equation is obtained. Copyright © 1999 John Wiley & Sons, Ltd.     

Sets of finite perimeter and the Hausdorff-Gauss measure on the Wiener space

In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinite-dimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff-Gauss measure restricted on the measure-theoretic boundary.     

A New Implicit Method for Surface Segmentation by Minimal Paths in 3D Images

We introduce a novel implicit approach for single-object segmentation in 3D images. The boundary surface of this object is assumed to contain two known curves (the constraining curves), given by an expert. The aim of our method is to find the wanted surface by exploiting as much as possible the information given in the supplied curves and in the image. As for active surfaces, we use a cost potential that penalizes image regions of low interest (most likely areas of low gradient or too far from the surface to be extracted). In order to avoid local minima, we introduce a new partial differential equation and use its solution for segmentation. We show that the zero level set of this solution contains the constraining curves as well as a set of paths joining them. We present a fast implementation that has been successfully applied to 3D medical and synthetic images.     

Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator

 We establish, by simple semigroup arguments, a Lévy–Gromov isoperimetric inequality for the invariant measure of an infinite dimensional diffusion generator of positive curvature with isoperimetric model the Gaussian measure. This produces in particular a new proof of the Gaussian isoperimetric inequality. This isoperimetric inequality strengthens the classical logarithmic Sobolev inequality in this context. A local version for the heat kernel measures is also proved, which may then be extended into an isoperimetric inequality for the Wiener measure on the paths of a Riemannian manifold with bounded Ricci curvature. Oblatum 19-VI-1995     

Solution of a matrix Wiener–Hopf equation connected with the plane wave diffraction by an impedance loaded parallel plate waveguide

A matrix Wiener–Hopf equation connected with a new canonical diffraction problem is solved explicitly. We consider the diffraction of a plane electromagnetic wave by an impedance loaded parallel plate waveguide formed by a two‐part impedance plane and a parallel perfectly conducting half‐plane. The representation of the solution to the boundary‐value problem in terms of Fourier integrals leads to a matrix Wiener–Hopf equation. The exact solution is obtained in terms of two infinite sets of unknown coefficients satisfying two infinite systems of linear algebraic equations. These systems are solved numerically and the influence of the parameters such as the waveguide spacing and the surface impedances of the two‐part plane on the diffraction phenomenon is shown graphically. Copyright © 2005 John Wiley & Sons, Ltd.     

On the expected surface area of the Wiener sausage

For parallel neighborhoods of the paths of the d ‐dimensional Brownian motion, so‐called Wiener sausages, formulae for the expected surface area are given for any dimension d ≥ 2. It is shown by means of geometric arguments that the expected surface area is equal to the first derivative of the mean volume of the Wiener sausage with respect to its radius (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Advanced MCMC methods for sampling on diffusion pathspace

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte-Carlo methods. We study here an advanced version of familiar Markov-chain Monte-Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate its computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinite-dimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.     

Note on generalized Wiener integrals

We establish a change of path formula for generalized Wiener integrals and develop a rotation theorem for Wiener measure with respect to Gaussian paths.