Measure of the multiple self-intersection set of a markov process

Let X(t), t ≧ 0, be a Markov process in R^m with homogeneous transition density p(t; x, y). For a closed bounded set B ? R^m, X is said to have a self-intersection of order r ≧ 2 in B if there are distinct points t₁ < … < t_r such that X(t₁) ∈ B and X(t_j) = X(t¹), for j = 2,…, r. The focus of this work is the Hausdorff measure, suitably defined, of the set of such r-tuples. The main result is that under general conditions on p(t; x, y) as well as the specific condition there is a measure function M(t), defined in terms of the integral above, such that the corresponding Hausdorff measure of self-intersection set is positive, with positive probability. The results are applied to Lévy and diffusion processes, and are shown to extend recent results in this area.     

The exact Hausdorff measures for the graph and image of a multidimensional iterated Brownian motion

Let {W(t),t∈R}, {B(t),t∈R } be two independent Brownian motions on R with W(0) = B(0) = 0. In this paper, we shall consider the exact Hausdorff measures for the image and graph sets of the d-dimensional iterated Brownian motion X(t), where X(t) = (Xi(t),... ,Xd(t)) and X1(t),... ,Xd(t) are d independent copies of Y(t) = W(B(t)). In particular, for any Borel set Q (?) (0,∞), the exact Hausdorff measures of the image X(Q) = {X(t) : t∈Q} and the graph GrX(Q) = {(t, X(t)) :t∈Q}are established.     

On the Vector Process Obtained by Iterated Integration of the Telegraph Signal

We analyse the vector process (X ₀(t), X ₁(t),...,X _n(t), t > 0) where , and X ₀(t) is the o two-valued telegraph process.In particular, the hyperbolic equations governing the joint distributions of the process are derived and analysed.Special care is given to the case of the process (X ₀(t), X ₁(t), X ₂(t), t > 0) representing a randomly accelerated motion where some explicit results on the probability distribution are derived.     

The square of a Gaussian Markov process and nonlinear prediction

An explicit formula is obtained for the nonlinear predictor of Y(t) = X(t)² − E(X(t)²), where X(t) is an N-ple Gaussian Markov process.     

PRV property and the <Emphasis Type="Italic">ϕ</Emphasis>-asymptotic behavior of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 445–465, October–December, 2007.     

Limit theorems for a class of diffusion processes

We consider a diffusion process {x(t)} on a compact Riemannian manifold with generator δ/2 + b. A current‐valued continuous stochastic process {X _t} in the sense of Itô [8] corresponds to {x(t)} by considering the stochastic line integral X _t(a) along {x(t)} for every smooth 1-form a. Furthermore {X _t} is decomposed into the martingale part and the bounded variation part as a current-valued continuous process. We show the central limit theorems for {X _t} and the martingale part of {X _t}. Occupation time laws for recurrent diffusions and homogenization problems of periodic diffusions are closely related to these theorems     

A critical case for Brownian slow points

Summary LetX _t be a Brownian motion and letS(c) be the set of realsr0 such that üX _r+t –X _r üct, 0th, for someh=h(r)>0. It is known thatS(c) is empty ifc<1 and nonempty ifc>1, a.s. In this paper we prove thatS(1) is empty a.s.This research was partially supported by NSF Grant 9322689.     

Limit theorems for supremum of Gaussian processes over a random interval

Let {X(t), t ≥ 0} be a centered stationary Gaussian process with correlation r(t)such that 1-r(t) is asymptotic to a regularly varying function. With T being a nonnegative random variable and independent of X(t), the exact asymptotics of P(sup_(t∈[0,T])X(t) x) is considered, as x →∞.     

The t-invariant of analytic set germs of dimension 2

Let be an analytic set germ of dimension 2. We study the invariant t(X₀) defined as the least integer t such that any open semianalytic set germ of X₀ can be written as a union of t basic open set germs. It is known that 2≤t(X₀)≤3. In this note we provide a geometric criterion to determine the exact value of t(X₀).     

Packing dimension and Ahlfors regularity of porous sets in metric spaces

Let X be a metric measure space with an s-regular measure μ. We prove that if A ì X{A\subset X} is r{\varrho} -porous, then dim_p(A) ￡ s-cr^s{{\rm {dim}_p}(A)\le s-c\varrho^s} where dim_p is the packing dimension and c is a positive constant which depends on s and the structure constants of μ. This is an analogue of a well known asymptotically sharp result in Euclidean spaces. We illustrate by an example that the corresponding result is not valid if μ is a doubling measure. However, in the doubling case we find a fixed N ì X{N\subset X} with μ(N) = 0 such that dim_p(A) ￡ dim_p(X)-c(log\tfrac1r)^-1r^t{{\rm {dim}_p}(A)\le{\rm {dim}_p}(X)-c(\log \tfrac1\varrho)^{-1}\varrho^t} for all r{\varrho} -porous sets A ì X\ N{A \subset X{\setminus} N} . Here c and t are constants which depend on the structure constant of μ. Finally, we characterize uniformly porous sets in complete s-regular metric spaces in terms of regular sets by verifying that A is uniformly porous if and only if there is t < s and a t-regular set F such that A ì F{A\subset F} .     

Decomposition of a class of functionals and the predictable representation theorem on Banach spaces

LetB be a separable real Banach space andX(t) be a symmetric conservative diffusion process taking values inB. In this paper, we decompose the functionalu(X(t),t) into a sum of a square integrable martingale and a regular 0-quadratic variation process. On this basis, we establish the predictable representation theorem ofX(t).This project is supported by the National Natural Science Foundation of China.     

Some path properties of generalized Lévy sheet

Let X(t) be an N parameter generalized Lévy sheet taking values in ℝ^d with a lower index α, ℜ = {(s, t] = ∏ _i=1 ^N (s _i, t _i], s _i < t _i}, E(x, Q) = {t ∈ Q: X(t) = x}, Q ∈ ℜ be the level set of X at x and X(Q) = {x: ∃t ∈ Q such that X(t) = x} be the image of X on Q. In this paper, the problems of the existence and increment size of the local times for X(t) are studied. In addition, the Hausdorff dimension of E(x, Q) and the upper bound of a uniform dimension for X(Q) are also established.     

On Osgood theorem in Banach spaces

Let X be a real Banach space, ω : [0, +∞) → ? be an increasing continuous function such that ω(0) = 0 and ω(t + s) ≤ ω(t) + ω(s) for all t, s ∈ [0, +∞). According to the infinite dimensional analog of the Osgood theorem if ∫¹₀ (ω(t))^?1 dt = ∞, then for any (t₀, x₀) ∈ ?×X and any continuous map f : ?×X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all t ∈ ?, x, y ∈ X, the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has a unique solution in a neighborhood of t₀. We prove that if X has a complemented subspace with an unconditional Schauder basis and ∫¹₀ (ω(t))^?1 dt < ∞ then there exists a continuous map f : ? × X → X such that ∥f(t, x) – f(t, y)∥ ≤ ω(∥x – y∥) for all (t, x, y) ∈ ? × X × X and the Cauchy problem (t) = f(t, x(t)), x(t₀) = x₀ has no solutions in any interval of the real line.     

No TVD Fields for 1-D Isentropic Gas Flow

We prove that isentropic gas flow does not admit non-degenerate TVD fields on any invariant set ?(r ₀, s ₀) = {r ₀ < r < s < s ₀}, where r, s are Riemann coordinates. A TVD field refers to a scalar field whose spatial variation Var _X (?(τ(t, X), u(t, X))) is non-increasing in time along entropic solutions. The result is established under the assumption that the Riemann problem defined by an overtaking shock-rarefaction interaction gives the asymptotic states in the exact solution.

Little is known about global existence of large-variation solutions to hyperbolic systems of conservation laws u _t  + f(u) _x  = 0. In particular it is not known if isentropic gas flow admits a priori BV bounds which apply to all BV data.

In the few cases where such results are available (scalar case, Temple class, systems satisfying Bakhvalov's condition, isothermal gas dynamics) there are TVD fields which play a key role for existence. Our results show that the same approach cannot work for isentropic flow.     

Asymptotic expansions in the conditional central limit theorem

Let X_n, n , be i.i.d. with mean 0, variance 1, and E(¦X_n¦^r) < ∞ for some r 3. Assume that Cramér's condition is fulfilled. We prove that the conditional probabilities P(1/√n Σ_{i = 1}ⁿ X_i t¦B) can be approximated by a modified Edgeworth expansion up to order o(1/n^{(r − 2)/2)}), if the distances of the set B from the σ-fields σ(X₁, …, X_n) are of order O(1/n^{(r − 2)/2)}(lg n)^β), where β < −(r − 2)/2 for r and β < −r/2 for r . An example shows that if we replace β < −(r − 2)/2 by β = −(r − 2)/2 for r (β < −r/2 by β = −r/2 for r ) we can only obtain the approximation order O(1/n^{(r − 2)/2)}) for r (O(lg lgn/n^{(r − 2)/2)}) for r ).     

Solving an Inverse First-Passage-Time Problem for Wiener Process Subject to Random Jumps from a Boundary

We study an inverse first-passage-time problem for Wiener process X(t) subject to random jumps from a boundary c. Let be given a threshold S > X(0); and a distribution function F on [0, + ∞). The problem consists of finding the distribution of the jumps which occur when X(t) hits c, so that the first-passage time of X(t) through S has distribution F.     

A bivariate Lévy process with negative binomial and gamma marginals

The joint distribution of X and N, where N has a geometric distribution and X is the sum of N IID exponential variables (independent of N), is infinitely divisible. This leads to a bivariate Lévy process {(X(t),N(t)),t≥0}, whose coordinates are correlated negative binomial and gamma processes. We derive basic properties of this process, including its covariance structure, representations, and stochastic self-similarity. We examine the joint distribution of (X(t),N(t)) at a fixed time t, along with the marginal and conditional distributions, joint integral transforms, moments, infinite divisibility, and stability with respect to random summation. We also discuss maximum likelihood estimation and simulation for this model.     

Multifunctions of faces for conditional expectations of selectors and Jensen's inequality

Let (T, , P) be a probability space, a P-complete sub-δ-algebra of and X a Banach space. Let multifunction t → Γ(t), t T, have a (X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε E_p x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X = , then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality.     

Non-linear prediction of the degree n of a Gaussian N-ple Markov process

An explicit formula is obtained for the nonlinear predicion of Y(t) = Xⁿ(t), where X(t) is an N-ple Gaussian Markov process.     

The order of the differentials in the Atiyah-Hirzebruch spectral sequence

LetF _* be the homology theory corresponding to a spectrumF and consider the Atiyah-Hirzebruch spectral sequenceE _s,t ² H _s (X; _t F) F _s+t (X) for a bounded below spectrum (or CW-complex)X. This paper shows that the images of the differentials d _s,t ^r :E _s,t ^r E _s ^{–r,t+r–1r} in this spectral sequence are always torsion groupsof finite exponent and that this exponent isbounded in a very universal way: we prove the existence of integersR _r forr2 such thatR _r d _s,t ^r =0 for any spectrumF, for any bounded below spectrumX and for all integersr2,s andt. The interesting point is that these upper boundsR _r for the additive order of the differentials d _s,t ^r dependonly onr, and that the result holds without any hypothesis on the spectrumF. In certain special cases, this implies that the spectral sequence collapses and even that the extension problems given by itsE -term are trivial.