Exit time and invariant measure asymptotics for small noise constrained diffusions

Constrained diffusions, with diffusion matrix scaled by small ?>0, in a convex polyhedral cone G⊂R^k, are considered. Under suitable stability assumptions small noise asymptotic properties of invariant measures and exit times from domains are studied. Let B⊂G be a bounded domain. Under conditions, an “exponential leveling” property that says that, as ?→0, the moments of functionals of exit location from B, corresponding to distinct initial conditions, coalesce asymptotically at an exponential rate, is established. It is shown that, with appropriate conditions, difference of moments of a typical exit time functional with a sub-logarithmic growth, for distinct initial conditions in suitable compact subsets of B, is asymptotically bounded. Furthermore, as initial conditions approach 0 at a rate ?² these moments are shown to asymptotically coalesce at an exponential rate.     

On the convergence of LePage series in Skorokhod space

We consider the problem of the convergence of the so-called LePage series in the Skorokhod space D^d=D([0,1],R^d) and provide a simple criterion based on the moments of the increments of the random process involved in the series. This provides a simple sufficient condition for the existence of an α-stable distribution on D^d with given spectral measure.     

Momentum estimates and ergodicity for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise

In this paper, we consider the ergodicity of invariant measures for the stochastic Ginzburg-Landau equation with degenerate random forcing. First, we show the existence and pathwise uniqueness of strong solutions with H¹-initial data, and then the existence of an invariant measure for the Feller semigroup by the Krylov-Bogoliubov method. Then in the case of degenerate additive noise, using the notion of asymptotically strong Feller property, we prove the uniqueness of invariant measures for the transition semigroup.     

On the stochastic Benjamin-Ono equation

We discuss the Cauchy problem for the stochastic Benjamin-Ono equation in the function class Hs(R), s>3/2. When there is a zero-order dissipation, we also establish the existence of an invariant measure with support in H²(R). Many authors have discussed the Cauchy problem for the deterministic Benjamin-Ono equation. But our results are new for the stochastic Benjamin-Ono equation. Our goal is to extend known results for the deterministic equation to the stochastic equation.     

Martingale representation for Poisson processes with applications to minimal variance hedging

We consider a Poisson process η on a measurable space equipped with a strict partial ordering, assumed to be total almost everywhere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with η), which was previously known only in the special case, when λ is the product of Lebesgue measure on R₊ and a σ-finite measure on another space X. Our proof is new and based on only a few basic properties of Poisson processes and stochastic integrals. We also consider the more general case of an independent random measure in the sense of Itô of pure jump type and show that the Clark-Ocone type representation leads to an explicit version of the Kunita-Watanabe decomposition of square integrable martingales. We also find the explicit minimal variance hedge in a quite general financial market driven by an independent random measure.     

Krengel-Lin decomposition for probability measures on hypergroups

A Markov operator P on a σ-finite measure space (X,Σ,m) with invariant measure m is said to have Krengel-Lin decomposition if L²(X)=E₀⊕L²(X,Σ_d) where E₀={f∈L²(X)∣‖Pⁿ(f)‖→0} and Σ_d is the deterministic σ-field of P. We consider convolution operators and we show that a measure λ on a hypergroup has Krengel-Lin decomposition if and only if the sequence converges to an idempotent or λ is scattered. We verify this condition for probabilities on Tortrat groups, on commutative hypergroups and on central hypergroups. We give a counter-example to show that the decomposition is not true for measures on discrete hypergroups.     

An intermediate Baum-Katz theorem

We extend the classical Hsu-Robbins-Erd?s theorem to the case when all moments exist, but the moment generating function does not, viz., we assume that Eexp{(log⁺|X|)^α}<∞ for some α>1. We also present multi-index versions of the same and of a related result due to Lanzinger in which the assumption is that Eexp{|X|^α}<∞ for some α∈(0,1).     

Geometric ergodicity of Harris recurrent Marcov chains with applications to renewal theory

Let (X_n) be a positive recurrent Harris chain on a general state space, with invariant probability measure π. We give necessary and sufficient conditions for the geometric convergence of λPⁿf towards its limit π(f), and show that when such convergence happens it is, in fact, uniform over f and in L¹(π)-norm. As a corollary we obtain that, when (Xn) is geometrically ergodic, ∝ π(dx)6Pⁿ(x,·)-π6 converges to zero geometrically fast. We also characterize the geometric ergodicity of (X_n) in terms of hitting time distributions. We show that here the so-called small sets act like individual points of a countable state space chain. We give a test function criterion for geometric ergodicity and apply it to random walks on the positive half line. We apply these results to non-singular renewal processes on [0,∞) providing a probabilistic approach to the exponencial convergence of renewal measures.     

Operator Aczél inequality

We establish several operator versions of the classical Aczél inequality. One of operator versions deals with the weighted operator geometric mean and another is related to the positive sesquilinear forms. Some applications including the unital positive linear maps on C^*-algebras and the unitarily invariant norms on matrices are presented.     

Absolute continuity of the invariant measure in piecewise deterministic Markov Processes having degenerate jumps

We consider piecewise deterministic Markov processes with degenerate transition kernels of the house-of-cards- type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the invariant measure of the process. Finally, we obtain finer results on the regularity of the one-dimensional marginals of the invariant measure, using integration by parts with respect to the jump times.     

Maximal dissipativity of Kolmogorov operators with Cahn-Hilliard type drift term

We prove the existence of invariant measures μ for Kolmogorov operators LF associated with semilinear stochastic partial differential equations with Cahn-Hilliard type drift term. Based on gradient estimates on the pseudo-resolvent associated with LF and a priori estimates for the moments of μ we prove maximal dissipativity of LF in the space L¹(μ).     

Harnack inequalities for Ornstein-Uhlenbeck processes driven by Lévy processes

By using the existing sharp estimates of the density function for rotationally invariant symmetric α-stable Lévy processes and rotationally invariant symmetric truncated α-stable Lévy processes, we obtain that the Harnack inequalities hold for rotationally invariant symmetric α-stable Lévy processes with α∈(0,2) and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric α-stable Lévy process, while the logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated α-stable Lévy processes.     

On the local time of random walk on the 2-dimensional comb

We study the path behaviour of general random walks, and that of their local times, on the 2-dimensional comb lattice C² that is obtained from Z² by removing all horizontal edges off the x-axis. We prove strong approximation results for such random walks and also for their local times. Concentrating mainly on the latter, we establish strong and weak limit theorems, including Strassen-type laws of the iterated logarithm, Hirsch-type laws, and weak convergence results in terms of functional convergence in distribution.     

Multiscale diffusion approximations for stochastic networks in heavy traffic

Stochastic networks with time varying arrival and service rates and routing structure are studied. Time variations are governed by, in addition to the state of the system, two independent finite state Markov processes X and Y. The transition times of X are significantly smaller than typical inter-arrival and processing times whereas the reverse is true for the Markov process Y. By introducing a suitable scaling parameter one can model such a system using a hierarchy of time scales. Diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. In particular, it is shown that, under certain conditions, properly normalized buffer content processes converge weakly to a reflected diffusion. The drift and diffusion coefficients of this limit model are functions of the state process, the invariant distribution of X, and a finite state Markov process which is independent of the driving Brownian motion.     

Asymptotics of the principal eigenvalue and expected hitting time for positive recurrent elliptic operators in a domain with a small puncture

Let X(t) be a positive recurrent diffusion process corresponding to an operator L on a domain D⊆R^d with oblique reflection at ∂D if D≠R^d. For each x∈D, we define a volume-preserving norm that depends on the diffusion matrix a(x). We calculate the asymptotic behavior as ε→0 of the expected hitting time of the ε-ball centered at x and of the principal eigenvalue for L in the exterior domain formed by deleting the ball, with the oblique derivative boundary condition at ∂D and the Dirichlet boundary condition on the boundary of the ball. This operator is non-self-adjoint in general. The behavior is described in terms of the invariant probability density at x and Det(a(x)). In the case of normally reflected Brownian motion, the results become isoperimetric-type equalities.     

FBSDEs with time delayed generators: L-solutions, differentiability, representation formulas and path regularity

We extend the work of Delong and Imkeller (2010) [6] and [7] concerning backward stochastic differential equations with time delayed generators (delay BSDEs). We give moment and a priori estimates in general L^p-spaces and provide sufficient conditions for the solution of a delay BSDE to exist in L^p. We introduce decoupled systems of SDEs and delay BSDEs (delay FBSDEs) and give sufficient conditions for their variational differentiability. We connect these variational derivatives to the Malliavin derivatives of delay FBSDEs via the usual representation formulas. We conclude with several path regularity results, in particular we extend the classic L²-path regularity to delay FBSDEs.     

Renewal theory with a trend

We prove some analogs of results from renewal theory for random walks in the case when there is a drift, more precisely when the mean of the kth summand equals k^γμ, k≥1, for some μ>0 and 0<γ≤1.     

An example on movable approximations of a minimal set in a continuous flow

In the present paper the study of flows on n-manifolds in particular in dimension three, e.g., R³, is motivated by the following question. Let A be a compact invariant set in a flow on X. Does every neighbourhood of A contain a movable invariant set M containing A? It is known that a stable solenoid in a flow on a 3-manifold has approximating periodic orbits in each of its neighbourhoods. The solenoid with the approximating orbits form a movable set, although the solenoid is not movable. Not many such examples are known. The main part of the paper consists of constructing an example of a set in R³ that is not stable, is not a solenoid, and is approximated by Denjoy-like invariant sets instead of periodic orbits. As in the case of a solenoid, the constructed set is an inverse limit of its approximating sets. This gives a partial answer to the above question.     

Time-dependent diffusion operators on <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>

We study the Cauchy problem for time-dependent diffusion operators with singular coefficients on L¹-spaces induced by infinitesimal invariant measures. We give sufficient conditions on the coefficients such that the Cauchy-Problem is well-posed. We construct associated diffusion processes with the help of the theory of generalized Dirichlet forms. We apply our results in particular to construct a large class of Nelson-diffusions that could not been constructed before.