Invariant measure of stochastic higher order KdV equation driven by Poisson processes

The current paper is devoted to stochastic damped KdV equations of higher order driven by Poisson process. We establish the well-posedness of stochastic damped higher-order KdV equations, and prove that there exists an unique invariant measure for deterministic initial conditions. Some discussion on the general pure jump noise case are also provided.

Unique Ergodicity for a Class of Stochastic Hyperbolic Equations with Additive Space-Time White Noise

In this paper, we consider a certain class of second order nonlinear PDEs with damping and space-time white noise forcing, posed on the d-dimensional torus. This class includes the wave equation for $$d=1$$ d = 1 and the beam equation for $$d\le 3$$ d ≤ 3 . We show that the Gibbs measure is the unique invariant measure for this system. Since the flow does not satisfy the strong Feller property, we introduce a new technique for showing unique ergodicity. This approach may be also useful in situations in which finite-time blowup is possible.

Exponential ergodicity for non-Lipschitz multivalued stochastic differential equations with Lévy jumps

We study the exponential ergodicity of diffusions generated by a multivalued stochastic differential equation with Lévy jumps when the coefficients are non-Lipschitz continuous by proving that the transition semigroup is strongly Feller and irreducible, and that it admits a unique invariant measure. This is obtained through an [Formula: see text]-convergence result, Girsanov’s theorem, coupling method combined and a stopping argument.

On some symmetric multidimensional continued fraction algorithms

We compute explicitly the density of the invariant measure for the reverse algorithm which is absolutely continuous with respect to Lebesgue measure, using a method proposed by Arnoux and Nogueira. We also apply the same method on the unsorted version of the Brun algorithm and Cassaigne algorithm. We illustrate some experimentations on the domain of the natural extension of those algorithms. For some other algorithms, which are known to have a unique invariant measure absolutely continuous with respect to Lebesgue measure, the invariant domain found by this method seems to have a fractal boundary, and it is unclear whether it is of positive measure.

APPROXIMATION BY ABSOLUTELY CONTINUOUS INVARIANT MEASURES OF ITERATED FUNCTION SYSTEMS WITH PLACE-DEPENDENT PROBABILITIES

Let [Formula: see text] be the attractor (fractal) of a contractive iterated function system (IFS) with place-dependent probabilities. An IFS with place-dependent probabilities is a random map [Formula: see text] where the probabilities [Formula: see text] of switching from one transformation to another are functions of positions, that is, at each step, the random map [Formula: see text] moves the point [Formula: see text] to [Formula: see text] with probability [Formula: see text]. If the random map [Formula: see text] has a unique invariant measure [Formula: see text], then the support of [Formula: see text] is the attractor [Formula: see text]. For a bounded region [Formula: see text], we prove the existence of a sequence [Formula: see text] of IFSs with place-dependent probabilities whose invariant measures [Formula: see text] are absolutely continuous with respect to Lebesgue measure. Moreover, if [Formula: see text] is a compact metric space, we prove that [Formula: see text] converges weakly to [Formula: see text] as [Formula: see text] We present examples with computations.

Invariant measure for the Vasicek interest rate model in the Heath–Jarrow–Morton–Musiela framework

In this paper we study a particular class of forward rate problems, related to the Vasicek model, where the driving equation is a linear Gaussian stochastic partial differential equation. We first give an existence and uniqueness results of the related mild solution in infinite dimensional setting, then we study the related Ornstein–Uhlenbeck semigroup with respect to the determination of a unique invariant measure for the associated Heath–Jarrow–Morton–Musiela model.