On ergodicity of some Markov processes

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak-$^*$ ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak-$^*$ mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.Comment: Published in at http://dx.doi.org/10.1214/09-AOP513 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Lévy–Ornstein–Uhlenbeck transition semigroup as second quantized operator

AbstractLet μ be an invariant measure for the transition semigroup (Pt) of the Markov family defined by the Ornstein–Uhlenbeck type equationdX=AXdt+dL on a Hilbert space E, driven by a Lévy process L. It is shown that for any t⩾0, Pt considered on L2(μ) is a second quantized operator on a Poisson Fock space of eAt. From this representation it follows that the transition semigroup corresponding to the equation on E=R, driven by an α-stable noise L, α∈(0,2), is neither compact nor symmetric

Cubature on Wiener space in infinite dimension

We prove a stochastic Taylor expansion for SPDEs and apply this result to obtain cubature methods, i. e. high order weak approximation schemes for SPDEs, in the spirit of T. Lyons and N. Victoir. We can prove a high-order weak convergence for well-defined classes of test functions if the process starts at sufficiently regular initial values. We can also derive analogous results in the presence of L\'evy processes of finite type, here the results seem to be new even in finite dimension. Several numerical examples are added.Comment: revised version, accepted for publication in Proceedings Roy. Soc.

Uniform large deviations for the nonlinear Schrodinger equation with multiplicative noise

Uniform large deviations for the laws of the paths of the solutions of the stochastic nonlinear Schrodinger equation when the noise converges to zero are presented. The noise is a real multiplicative Gaussian noise. It is white in time and colored in space. The path space considered allows blow-up and is endowed with a topology analogue to a projective limit topology. Thus a large variety of large deviation principle may be deduced by contraction. As a consequence, asymptotics of the tails of the law of the blow-up time when the noise converges to zero are obtained

Metaphorical Effects in the Works of Annie Ernaux

While writing her fourth book, La Place (1984), Ernaux abandoned the genre of the novel and adopted a new prose style that was devoid of metaphor, and other hallmarks of literary writing in favor of a "flat" style. In this study, I show that Ernaux's writing is not as "flat" as it appears to be, and that the author has been maneuvering around her ambivalence to metaphor--and its strong association with literary style--for a long time. An attentive reading, as I have illustrated, reveals new dimensions in her writing and opens up her works to fresh interpretations. An appreciation for the evolution of her style, and the artistic effects hidden below her écriture plate, requires, however, a familiarity with her oeuvre as a whole and active reflection on the reader's part. This dissertation emphasizes Ernaux's approaches to metaphor throughout a body of work that now spans four decades

The investor problem based on the HJM model

We consider a consumption-investment problem (both on finite and infinite time horizon) in which the investor has an access to the bond market. In our approach prices of bonds with different maturities are described by the general HJM factor model. We assume that the bond market consists of entire family of rolling bonds and the investment strategy is a general signed measure distributed on all real numbers representing time to maturity specifications for different rolling bonds. In particular, we can consider portfolio of coupon bonds. The investor's objective is to maximize time-additive utility of the consumption process. We solve the problem by means of the HJB equation for which we prove required regularity of its solution and all required estimates to ensure applicability of the verification theorem. Explicit calculations for affine models are presented.Comment: v2 - 26 pages, detailed calculations of G2++ model, extended proof of theorem 4.1, two references added( [2] and [33]), v3 - 28 pages, revised version after reviews, (v4) - 30 pages, language corrections, (v5),(v6) - 29 pages, final correction

Time irregularity of generalized Ornstein--Uhlenbeck processes

The paper is concerned with the properties of solutions to linear evolution equation perturbed by cylindrical L\'evy processes. It turns out that solutions, under rather weak requirements, do not have c\`adl\`ag modification. Some natural open questions are also stated