Recent Developments in Stochastic Analysis and Related Topics

Preface

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We give precise Gaussian upper and lower bound estimates on heat kernels on Riemannian manifolds with poles under assumptions that the Riemannian curvature tensor goes to 0 sufficiently fast at infinity. Under additional assumptions on the curvature, we give estimates on the logarithmic derivatives of the heat kernels. The proof relies on the Elworthy-Truman’s formula of heat kernels and Elworthy and Yor’s observation on the derivative process of certain stochastic flows. As an application of them, we prove logarithmic Sobolev inequalities on pinned path spaces over such Riemannian manifolds.

We study the equivalence of bipartite mixed states under local unitary transformations. The nonlocal properties of generic bipartite quantum states in C^M ⊗ C^N composite systems are investigated. A complete set of invariants under local unitary transformations for these states is presented. It is shown that two generic density matrices are locally equivalent if and only if all these invariants have equal values in these density matrices.

Invariant quantities of the classical motion of an ideal incompressible fluid in a two-dimensional bounded domain are used to construct a family {Π_α}_α of probability measures of the Gibbs form, which are invariant under the flow. The Gibbs exponent H is given by the renormalized energy. These measures are supported by the space of configuration Γ, i.e. the fluid vorticity is concentrated in a finite number of distinct points. Properties of a deterministic vortex dynamics having Π_α as invariant measure are investigated; in particular Markov uniqueness is proven. The classical (pre-)Dirichlet form associated to Π_α is also introduced and analyzed.

We study the limiting distribution of the sum as t → ∞, N → ∞, where (X_i) are i.i.d. random variables. Attention to such exponential sums has been motivated by various problems in random media theory. Examples include the quenched mean population size of a colony of branching processes with random branching rates and the partition function of Derrida’s Random Energy Model. In this paper, the problem is considered under the assumption that the log-tail distribution function h(x) = − log P{X_i > x} is regularly varying at infinity with index 1 < ϱ < ∞. An appropriate scale for the growth of N relative to t is of the form e^λH₀(t), where the rate function H₀(t) is a certain asymptotic version of the cumulant generating function H(t) = log E[e^tXi] provided by Kasahara’s exponential Tauberian theorem. We have found two critical points, 0< λ₁< λ₂ < ∞, below which the Law of Large Numbers and the Central Limit Theorem, respectively, break down. Below λ₂, we impose a slightly stronger condition of normalized regular variation of h. The limit laws here appear to be stable, with characteristic exponent α = α(ϱ, λ) ranging from 0 to 2 and with skewness parameter β = 1. A limit theorem for the maximal value of the sample {e^tXi, i = 1, ..., N} is also proved.

Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of unique solutions. We obtain a gap of regularity between the unique local and the global solutions, which are necessary to define statistical quantities like mean interface width or correlation functions.

This paper is concerned with the large time behaviour of solutions to the 2D stochastic Navier-Stokes equation on some unbounded domains. A notion of asymptotic compactness of a Random Dynamical System on a Polish space, introduced by authors in [11], is recalled. We present some results from that paper as well as announce some results from [12]. Firstly, we show that an asymptotically compact RDS on a separable Banach space has a non-empty compact Ω-limit set. Secondly, we show that the RDS generated by a stochastic NSEs on a 2D domain satisfying the Poincaré inequality is asymptotically compact on the space H of divergence free vector fields on D with finite L² norm. Thirdly, we show that this is also true for the space V of divergence free vector fields on D with finite H^1,2 norm. The first two results have been proved in [11], the third comes from the work in preparation [12]. This research is motivated by papers [35] by Rosa and [31] by Ju, where similar questions for dynamical systems generated by 2D deterministic NSEs were studied. One consequence of our main results is the existence of a Feller invariant measures (however the uniqueness of invariant measures remains an open problem). The case of stochastic NSEs in bounded domains have been studied earlier in [4], [8] and [38].

Some stochastic models of economic optimization are discussed in this paper. Due to the value in practice, the models are quite attractive. But our knowledge on them is still very limited, some fundamental problems remain open.

We begin with a short review of the study on some global economic models (or economy in large scale), the well-known input-output method and L. K. Hua’s fundamental theorem for the stability of economy. Then, we show that it is necessary to study the stochastic models. A collapse theorem for a non-controlling stochastic economic system is introduced. In the analysis of the system, the products of random matrices play a critical role. Especially, the first eigenvalue, the corresponding eigenfunctions and an ergodic theorem of Markov chains play a nice role here. Partial proofs are included. Some challenge open problems are also mentioned.

Suppose D is a bounded C^1,1 domain in Rⁿ and G_D is the Green function of the killed symmetric α-stable process in D. In this note we establish a sharp upper bound estimate for G_D in such a way that the explicit dependence of the constant in the estimate on α is also given, This sharp estimate allows us to recover the sharp upper bound estimate for the Green function of killed Brownian motion in D by letting α ↑ 2.

No abstract received.

By using a measure transformation method, the essential spectrum of the Laplacian in a noncompact Riemannian manifold is studied. The principle result given includes the corresponding main theorems in both Kumura [16] and Donnelly [9]. The essential spectrum of the Laplacian on one-forms is also studied.

The definition and the basic theory of Lévy processes on real Lie algebras is summarized and several examples are presented. It is also shown how KMS or temperature states can be constructed for such processes.

Generalized second order differential operators of the form are considered, where μ is a finite atomless Borel measure which is compactly supported on the real line. In the special case of self-similar measures – Hausdorff measures or, more general, self-similar measures with arbitrary weights –, the asymptotic distribution of the zeros of the eigenfunctions is determined.

Random field with paths given as restrictions of holomorphic functions to Euclidean space-time can be Wick-rotated by pathwise analytic continuation. Euclidean symmetries of the correlation functions then go over to relativistic symmetries. As a concrete example, convoluted point processes with interactions motivated from quantum field theory are discussed. A general scheme for the construction of Euclidean invariant infinite volume measures for systems of continuous particles with ferromagnetic interaction is given and applied to the models under consideration. Connections with Euclidean quantum field theory, Widom-Rowlinson and Potts models are pointed out. For the given models, pathwise analytic continuation and analytically continued correlation functions are shown to exist and to expose relativistic symmetries.

In this paper we present a general white noise approach to interacting quantum field theory by Hamiltonian strategy. Applying it into two models-P(ϕ)₂ and quantum fields we prove the existence as generalized operators under an appropriate framework of white noise analysis. In addition, some dynamical properties of quantum fields are given.

The smoothness of marginal distributions of solutions to SDE’s with jumps is investigated via the Malliavin calculus on the càd-làg space. A key lemma on the estimation of some functionals of semi-martingales is presented. Using the key lemma, the Hörmander theorem on the hypo-ellipticity of parabolic differential operators is generalized to a theorem for the integro-differential operators associated with the SDE’s. This article is essentially a summary of [3]. But the last section where the main theorem is presented is an improvement of [3].

No abstract received.

No abstract received.

We give central limit theorems for generalized set-valued random variables whose level sets are compact both in ℝ^d or in a Banach space under milder conditions than those obtained recently by the last two authors.

Let M be a complete Riemannian manifold, Δ be the Laplace-Beltrami operator, □ be the de Rham-Hodge operator, ∇ be the gradient operator, Ric be the Ricci curvature. Denote K(x) := inf{(Ric(x)υ, υ) : υ ∈ T_xM, ||υ|| = 1}, ∀ x ∈ M. Let .

In this paper we prove that if λ_∞(Δ/2 + K) ≥ 0, then the Littlewood-Paley-Stein function is bounded in L^p for all p ≥ 2; if λ_∞(Δ/2 + K) > 0, then the Littlewood-Paley-Stein function on one-forms is bounded in L^q for all 1 < q ≤ 2. As a corollary, the Riesz transform R₀(∆) := ∇∆^−1/2 is bounded in L^p for all p ≥ 2 provided that λ_∞(Δ/2 + K) > 0. Moreover, if λ_∞(Δ/2 + K) > -∞, we prove that the Riesz transform R_a(∆) := ∇(a+∆)^−1/2 is bounded in L^p for all p ≥ 2 and a > max{−λ_∞(Δ/2+K), 0}. Finally, we give a class of complete Riemannian manifolds on which the Ricci curvature is not uniformly bounded from below while the Riesz transform R_a(∆) is bounded in L^p for all a ≥ 0 and p ≥ 2.

Based on recent work, we discuss how the problem of infrared divergence in Nelson’s massless scalar field model can be tackled. We translate the problem originally formulated in terms of operators into a problem of stochastics by associating well specified processes to the particle and field operators. The ground state will be put in direct relationship with data given by Gibbs measures relative to these processes. When interpreted back, it turns out that in three dimensions the Nelson Hamiltonian has no ground state in Fock space and thus is not unitary equivalent with the Hamiltonian obtained from Euclidean quantization. In contrast, for dimensions higher than three the Nelson Hamiltonian does have a unique ground state in Fock space and the two Hamiltonians are unitary equivalent. We show how another representation for the 3D case can be constructed which eliminates infrared problems altogether.

An estimation problem is considered for a stochastic parabolic equation with an unknown random coefficient. The additional randomness in the coefficient generalizes a popular estimation problem that has been extensively studied in recent years. The filter estimate of the coefficient is constructed from a finite-dimensional projection of the solution of the equation. Under certain conditions this estimate is approximated using a generalized Kalman-Bucy filter whose filter variance tends to zero as the dimension of the projection increases.

A new interacting free Fock (IFF) space is studied in the present note. We show some general properties of the IFF structure and calculate the distributions of forward-backward field operators with respect to the vacuum state.

No abstract received.

In this paper we study the stochastic transport equation of convolution type. For general initial condition and its coefficients we give an explicit solution which is a well defined generalized stochastic process in a suitable distribution space. Under certain assumptions on the coefficients we also write the obtained solution as a convergent series of integrals.

In this article forward growth rates for the asset value of a company are discussed and connections between the Heath-Jarrow-Morton (HJM) and the Black-Scholes (BS) models are exhibited. A financial market model where forward growths rates for assets are modelled by Levy processes is also discussed. Practical results are obtained in a special case, in particular concerning the pricing of options and calibration of parameters.

Starting from elementary considerations about independence and Markov processes in classical probability we arrive at the new concept of conditional monotone independence (or operator-valued monotone independence). With the help of product systems of Hilbert modules we show that monotone conditional independence arises naturally in dilation theory.

We consider two different ways to force Brownian motion to be close to a submanifold of a riemannian manifold. We investigate their relationship and consider an application to the quantum mechanics of thin layers.