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This paper studies a variational formulation of the image matching problem. We consider a scenario in which a canonical representative image T is to be carried via a smooth change of variable into an image which is intended to provide a good fit to the observed data. The images are all defined on a compact set G ae IR 3 . The changes of variable are determined as solutions of the nonlinear Eulerian transport equation dj(s; x) ds = v(j(s; x); s); j(ø ; x) = x; (0:1) with the location j(0; x) in the canonical image carried to the location x in the deformed image. The variational problem then takes the form arg min v kvk 2 + Z G jT ffi j(0; x) \Gamma D(x)j 2 dx ; (0:2) where kvk is an appropriate norm on the velocity field v(\Delta; \Delta), and the second term attempts to enforce fidelity to the data. In this paper we derive conditions under which the variational problem described above is well posed. The key issue is the choice of the norm. Conditions are formulated u...

Geometric convergence of Markov chains in discrete time on a general state has been studied in detail in [15]. Here we develop a similar theory for '-irreducible continuous time processes, and consider the following types of criteria for geometric convergence: (a) the existence of exponentially bounded hitting times on one and then all suitably "small" sets; (b) the existence of "Foster-Lyapunov" or "drift" conditions for any one and then all skeleton and resolvent chains; (c) the existence of drift conditions on the extended generator e A of the process. We use the identity e AR fi = fi(R fi \Gamma I) connecting the extended generator and the resolvent kernels R fi , to show that, under a suitable aperiodicity assumption, exponential convergence is completely equivalent to any of (a)--(c). These conditions yield criteria for exponential convergence of unbounded as well as bounded functions of the chain. They enable us to identify the dependence of the convergence on the initial state ...

When a signal is emitted from a source, recorded by an array of transducers, timereversed, and re-emitted into the medium, it will refocus approximately on the source location. We analyze the refocusing resolution in a high frequency remote-sensing regime and show that, because of multiple scattering in an inhomogeneous or random medium, it can improve beyond the diffraction limit. We also show that the back-propagated signal from a spatially localized narrow-band source is self-averaging, or statistically stable, and relate this to the self-averaging properties of functionals of the Wigner distribution in phase space. Time reversal from spatially distributed sources is self-averaging only for broad-band signals. The array of transducers operates in a remote-sensing regime, so we analyze time reversal with the parabolic or paraxial wave equation.

. We consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing. We do this by proving the robustness of the spectra of their Perron-Frobenius operators. Introduction Let f : M !M be a dynamical system preserving some natural probability measure ¯ 0 with density ae 0 . This paper is motivated by the following question: does exponential mixing imply stochastic stability? Roughly speaking, exponential mixing of (f; ¯ 0 ) means that, for two observables ' and / on M , the correlation between ' ffi f n and / decays exponentially fast with n. Stochastic stability means that, if we add a small amount of random noise to f , obtaining at noise level ffl a Markov process with invariant density ae ffl , then ae ffl tends to ae 0 as ffl tends to zero. The following heuristic argument suggests an affirmative answer to this question. Consider the Perron-Frobenius operator L associated with f acting on ...

Invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems. The theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems, and for stochastic ordinary differential equations is relatively mature. In this paper, we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations. We first introduce a random graph transform and a fixed point theorem for non-autonomous systems. Then we show the existence of generalized fixed points which give the desired invariant manifolds.

We formulate and prove a Local Stable Manifold Theorem for stochastic differential equations (sde's) that are driven by spatial Kunita-type semimartingales with stationary ergodic increments. Both Stratonovich and It^o-type equations are treated. Starting with the existence of a stochastic flow for a sde, we introduce the notion of a hyperbolic stationary trajectory. We prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution. For Stratonovich sde's, the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde. The proof of the stable manifold theorem is based on Ruelle-Oseledec multiplicative ergodic theory.

In this paper we consider a #-irreducible continuous parameter Markov process # whose state space is a general topological space. The recurrence and Harris recurrence structure of # is developed in terms of generalized forms of resolvent chains, where we allow statemodulated resolvents and embedded chains with arbitrary sampling distributions. We show that the recurrence behavior of such generalized resolvents classifies the behavior of the continuous time process; from this we prove that hitting times on the small sets of a generalized resolvent chain provide criteria for, successively, (i) Harris recurrence of # (ii) the existence of an invariant probability measure # (or positive Harris recurrence of #) and (iii) the finiteness of #(f) for arbitrary f.

this article is to introduce the reader to certain aspects of stochastic differential systems, whose evolution depends on the past history of the state. Chapter I begins with simple motivating examples. These include the noisy feedback loop, the logistic time-lag model with Gaussian noise, and the classical "heat-bath" model of R. Kubo, modeling the motion of a "large" molecule in a viscous fluid. These examples are embedded in a general class of stochastic functional differential equations (sfde's). We then establish pathwise existence and uniqueness of solutions to these classes of sfde's under local Lipschitz and linear growth hypotheses on the coefficients. It is interesting to note that the above class of sfde's is not covered by classical results of Protter, Metivier and Pellaumail and Doleans-Dade. In Chapter II, we prove that the Markov (Feller) property holds for the trajectory random field of a sfde. The trajectory Markov semigroup is not strongly continuous for positive delays, and its domain of strong continuity does not contain tame (or cylinder) functions with evaluations away from 0. To overcome this difficulty, we introduce a class of quasitame functions. These belong to the domain of the weak infinitesimal generator, are weakly dense in the underlying space of continuous functions and generate the Borel

This paper concerns the fluid dynamics modelled by the stochastic flow