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We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method. ISSN:1615-3375 ISSN:1615-3383

We consider the global minimization of a polynomial on a compact set $$\mathbf {B}$$ . We show that each step of the Moment-SOS hierarchy has a nice and simple interpretation that complements the usual one. Namely, it computes coefficients of a polynomial in an orthonormal basis of $$L^2(\mathbf {B},\mu )$$ where $$\mu $$ is an arbitrary reference measure whose support is exactly $$\mathbf {B}$$ . The resulting polynomial is a certain density (with respect to $$\mu $$ ) of some signed measure on $$\mathbf {B}$$ . When some relaxation is exact (which generically takes place) the coefficients of the optimal polynomial density are values of orthonormal polynomials at the global minimizer and the optimal (signed) density is simply related to the Christoffel–Darboux (CD) kernel and the Christoffel function associated with $$\mu $$ . In contrast to the hierarchy of upper bounds which computes positive densities, the global optimum can be achieved exactly as integration against a polynomial (signed) density because the CD-kernel is a reproducing kernel, and so can mimic a Dirac measure (as long as finitely many moments are concerned).

In this article, we give some reviews concerning negative probabilities model and quasi-infinitely divisible at the beginning. We next extend Feller's characterization of discrete infinitely divisible distributions to signed discrete infinitely divisible distributions, which are discrete pseudo compound Poisson (DPCP) distributions with connections to the L\'evy-Wiener theorem. This is a special case of an open problem which is proposed by Sato(2014), Chaumont and Yor(2012). An analogous result involving characteristic functions is shown for signed integer-valued infinitely divisible distributions. We show that many distributions are DPCP by the non-zero p.g.f. property, such as the mixed Poisson distribution and fractional Poisson process. DPCP has some bizarre properties, and one is that the parameter $\lambda $ in the DPCP class cannot be arbitrarily small., Comment: Accepted for publication in Studia Scientiarum Mathematicarum, 10 October 2016

Let $a$ be a finite signed measure on $[-r, 0]$ with $r \in (0, \infty)$. Consider a stochastic process $(X^{(\vartheta)}(t))_{t\in[-r,\infty)}$ given by a linear stochastic delay differential equation \[ \mathrm{d} X^{(\vartheta)}(t) = \vartheta \int_{[-r,0]} X^{(\vartheta)}(t + u) \, a(\mathrm{d} u) \, \mathrm{d} t + \mathrm{d} W(t) , \qquad t \ge 0, \] where $\vartheta \in \mathbb{R}$ is a parameter and $(W(t))_{t\ge 0}$ is a standard Wiener process. Consider a point $\vartheta \in \mathbb{R}$, where this model is unstable in the sense that it is locally asymptotically Brownian functional with certain scalings $(r_{\vartheta,T})_{T\in(0,\infty)}$ satisfying $r_{\vartheta,T} \to 0$ as $T \to \infty$. A family $\{(X^{(\vartheta_T)}(t))_{t\in[-r,T]} : T \in (0, \infty)\}$ is said to be nearly unstable as $T \to \infty$ if $\vartheta_T \to \vartheta$ as $T \to \infty$. For every $\alpha \in \mathbb{R}$, we prove convergence of the likelihood ratio processes of the nearly unstable families $\{(X^{(\vartheta+\alpha \ r_{\vartheta,T})}(t))_{t\in[-r,T]}: T \in (0, \infty)\}$ as $T \to \infty$. As a consequence, we obtain weak convergence of the maximum likelihood estimator $\hat{\alpha}_T$ of $\alpha$ based on the observations $(X^{(\vartheta+\alpha \ r_{\vartheta,T})}(t))_{t\in[-r,T]}$ as $T \to \infty$. It turns out that the limit distribution of $\hat{\alpha}_T$ as $T \to \infty$ can be represented as the maximum likelihood estimator of a parameter of a process satisfying a stochastic differential equation without time delay., Comment: 15 pages

This article investigates the support detection problem using the LASSO estimator in the space of measures. More precisely, we study the recovery of a discrete measure (spike train) from few noisy observations (Fourier samples, moments...) using an $\ell_{1}$-regularization procedure. In particular, we provide an explicit quantitative localization of the spikes., Revised version, minor changes, 16pages

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process $ X_t = X_0 + \int_0^t b(X_s) \, {\rm d} B_s + \int_{\bf R} L^X(t,y)$ where $b$ is a measurable real function, $B$ is a Wiener process, and $\nu$ denotes a set function which is defined on the bounded Borel sets of the real line ${\bf R}$ such that it is a finite signed measure on ${\cal B}([-N,N])$ for every $N \in {\bf N}$. This kind of equation is, in dependence of using the right, the left, or the symmetric local time, usually studied under the atom condition $\nu(\{x\}) -1/2$, and $|\nu(\{x\})| < 1$, respectively. This condition allows us to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from corresponding results for equations without drift. The main aim of the present paper is to treat the cases $\nu(\{x\}) \ge 1/2$, $\nu(\{x\}) \le -1/2$, and $|\...

Suppose d ≥ 2 and α ∈ (1, 2). Let D be a (not necessarily bounded) C1,1 open set in ℝd and μ = (μ1, . . . , μd) where each μj is a signed measure on ℝd belonging to a certain Kato class of the rotationally symmetric α-stable process X. Let Xμ be an α-stable process with drift μ in ℝd and let Xμ,D be the subprocess of Xμ in D. In this paper, we derive sharp two-sided estimates for the transition density of Xμ,D.

In this paper, the properties of the super quantum measures are studied. Firstly, the products of Dirac measures are discussed; Secondly, based on the properties of Dirac measures, the structures of super quantum measures are characterized; At last, we prove that any super quantum measure can determine a unique diagonally positive strongly symmetric signed measure. This result verifies the conjecture which was proposed by Gudder.

In Chap. 5 we defined the Lebesgue integral of functions defined on \(\mathbb{R}\). In this chapter we show that the theory remains valid in a much more general framework;moreover, almost all proofs can be repeated word for word. The results of this chapter include integrals of several variables and integrals on probability spaces

In this paper, we establish sharp two-sided estimates for the Dirichlet heat kernels of a large class of time-dependent parabolic operators with singular drifts in C 1 , α -domain in R d , where d ⩾ 1 and α ∈ ( 0 , 1 ] . Our operator is L + μ ⋅ ∇ x , where L is a time-dependent uniformly elliptic divergent operator with Dini continuous coefficients and μ is a signed measure on ( 0 , ∞ ) × R d belonging to parabolic Kato class. Along the way, a gradient estimate is also established. Our method employs a combination of partial differential equations and perturbation techniques.

Assume that we observe a stationary Gaussian process X(t), $${t \in [-r, T]}$$ , which satisfies the affine stochastic delay differential equation $$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$$ where W(t), t ≥ 0, is a standard Wiener process independent of X(t), $${t\in [-r, 0]}$$ , and $${a_\vartheta}$$ is a finite signed measure on [−r, 0], $${\vartheta\in\Theta}$$ . The parameter $${\vartheta}$$ is unknown and has to be estimated based on the observation. In this paper we consider the case where $${\Theta=(\vartheta_0,\vartheta_1)}$$ , $${-\infty\

Given $\mathcal{X}$ , some measurable subset of Euclidean space, one sometimes wants to construct a finite set of points, $\mathcal{P}\subset\mathcal {X}$ , called a design, with a small energy or discrepancy. Here it is shown that these two measures of design quality are equivalent when they are defined via positive definite kernels $K:\mathcal{X}^{2}(=\mathcal{X}\times\mathcal {X})\to\mathbb{R}$ . The error of approximating the integral $\int_{\mathcal{X}}f(\boldsymbol{x})\,\mathrm{d}\mu(\boldsymbol{x})$ by the sample average of f over $\mathcal{P}$ has a tight upper bound in terms of the energy or discrepancy of $\mathcal{P}$ . The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the measure μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. If $\mathcal{X}$ is the orbit of a compact, possibly non-Abelian group, $\mathcal{G}$ , acting as measurable transformations of $\mathcal{X}$ and the kernel K is invariant under the group action, then it is shown that the equilibrium measure is the normalized measure on $\mathcal{X}$ induced by Haar measure on $\mathcal{G}$ . This allows us to calculate explicit representations of equilibrium measures.

Using Schauder's fixed point theorem, with the help of an integral representation in ‘Sharp conditions for weighted 1-dimensional Poincare inequalities’, Indiana Univ. Math. J., 49 (2000) 143–175, by Chua and Wheeden, we obtain existence and uniqueness theorems and ‘continuous dependence of average condition’ for average value problem: y ′ = F ( x , y ) , ∫ a b y ( x ) d v = y 0 where v is any probability measure on [ a , b ] under the usual conditions for initial value problem. We also extend our existence and uniqueness theorems in the case where v is just a signed measure with v [ a , b ] ≠ 0 and F : F ⊂ C [ a , b ] → L 1 [ a , b ] is a continuous operator .

In this paper we consider linear filtering for discontinuous processes determined by stochastic differential equations on a Hilbert space driven by signed measures in addition to Brownian motion. The dynamics of the observed data is governed by a differential equation driven by a square integrable martingale (not necessarily continuous) while perturbed by a signed measure. We formulate the filtering problem as an optimization problem on the space of bounded linear operator valued functions and present necessary and sufficient conditions for optimality. Further, we prove, under the assumption of finite dimensionality of the output space, that a Kalman-like filter exists and it is explicitly determined by a Riccati type evolution equation.

We study the Schrodinger equation (q − ℒ)u + µu = f, where ℒ is the generator of a Borel right process and µ is a signed measure on the state space. We prove the existence and uniqueness results in Lp, 1 ⩽ p < ∞. Since we consider measures µ charging no polar set, we have to use new tools: the Revuz formula with fine versions and the appropriate Revuz correspondence, the perturbation (subordination) operators (in the sense of G. Mokobodzki) induced by the regular strongly supermedian kernels. We extend the results on the Schrodinger equation to the case of a strongly continuous sub-Markovian resolvent of contractions on Lp. If the measure µ is positive then the perturbed process solves the martingale problem for ℒ − µ and its transition semigroup is given by the Feynman-Kac formula associated with the left continuous additive functional having µ as Revuz measure.

(A) The celebrated Gaussian quadrature formula on finite intervals tells us that the Gauss nodes are the zeros of the unique solution of an extremal problem. We announce recent results of Damelin, Grabner, Levesley, Ragozin and Sun which derive quadrature estimates on compact, homogenous manifolds embedded in Euclidean spaces, via energy functionals associated with a class of group-invariant kernels which are generalizations of zonal kernels on the spheres or radial kernels in euclidean spaces. Our results apply, in particular, to weighted Riesz kernels defined on spheres and certain projective spaces. Our energy functionals describe both uniform and perturbed uniform distribution of quadrature point sets. (B) Given \(\mathcal{X}\), some measurable subset of Euclidean space, one sometimes wants to construct, a design, a finite set of points, \(\mathcal{P} \subset \mathcal{X}\), with a small energy or discrepancy. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that these two measures of quality are equivalent when they are defined via positive definite kernels \(K:\mathcal{X}^2(=\mathcal{X}\times\mathcal{X}) \to \mathbb{R}\). The error of approximating the integral \(\int_{\mathcal{X}} f(\mathbf{\mathit{x}}) \, {\rm d} \mu(\mathbf{\mathit{x}})\) by the sample average of f over \(\mathcal{P}\) has a tight upper bound in terms the energy or discrepancy of \(\mathcal{P}\). The tightness of this error bound follows by requiring f to lie in the Hilbert space with reproducing kernel K. The theory presented here provides an interpretation of the best design for numerical integration as one with minimum energy, provided that the μ defining the integration problem is the equilibrium measure or charge distribution corresponding to the energy kernel, K. (C) Let \(\mathcal{X}\) be the orbit of a compact, possibly non Abelian group, \(\mathcal{G}\), acting as measurable transformations of \(\mathcal{X}\) and the kernel K is invariant under the group action. We announce recent results of Damelin, Hickernell, Ragozin and Zeng which show that the equilibrium measure is the normalized measure on \(\mathcal{X}\) induced by Haar measure on \(\mathcal{G}\). This allows us to calculate explicit representations of equilibrium measures. There is an extensive literature on the topics (A–C). We emphasize that this paper surveys recent work of Damelin, Grabner, Levesley, Hickernell, Ragozin, Sun and Zeng and does not mean to serve as a comprehensive survey of all recent work covered by the topics (A–C).

For a (non-symmetric) strong Markov process $X$, consider the Feynman--Kac semigroup \[T_t^Af(x):=\mathbb {E}^x\bigl[e^{A_t}f(X_t)\bigr],\quad x\in {\mathbb {R}^n}, t>0,\] where $A$ is a continuous additive functional of $X$ associated with some signed measure. Under the assumption that $X$ admits a transition probability density that possesses upper and lower bounds of certain type, we show that the kernel corresponding to $T_t^A$ possesses the density $p_t^A(x,y)$ with respect to the Lebesgue measure and construct upper and lower bounds for $p_t^A(x,y)$. Some examples are provided., Published at http://dx.doi.org/10.15559/15-VMSTA26 in the Modern Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA) by VTeX (http://www.vtex.lt/)

Let (Φ t ) t∈ℝ+ be a Harris ergodic continuous-time Markov process on a general state space, with invariant probability measure π. We investigate the rates of convergence of the transition function P t (x, ·) to π; specifically, we find conditions under which r(t)||P t (x, ·) − π|| → 0 as t → ∞, for suitable subgeometric rate functions r(t), where ||·|| denotes the usual total variation norm for a signed measure. We derive sufficient conditions for the convergence to hold, in terms of the existence of suitable points on which the first hitting time moments are bounded. In particular, for stochastically ordered Markov processes, explicit bounds on subgeometric rates of convergence are obtained. These results are illustrated in several examples.

This work provides some results on the convolutions of a special kind of signed measure known as the Poisson signed measure. We show a simple way of obtaining asymptotic decompositions into convolutions of Poisson signed measure that is appropriate for a broad range of lattice distributions. The work demonstrates the analogy between such decompositions and the classic Gram--Charlier type A and B decompositions and the Edgeworth-Cramer series.

We present an upper bound for the total variation distance between the generalized polynomial distribution and a finite signed measure, which is the convolution of two finite signed measures, one of which is of Kornya--Presman type. In the one-dimensional Poisson case, such a finite signed measure was first considered by K. Borovkov and D. Pfeifer [{\em J. Appl.\ Probab.}, 33 (1996), pp. 146--155].We give asymptotic relations in the one-dimensional case, and, as an example, the independent identically distributed record model is investigated.It turns out that here the approximation is of order $O(n^{-s}(\ln n)^{-{(s+1)/2}})$ for s being a fixed positive integer, whereas in the approximation with simple Kornya--Presman signed measures, we only have the rate $O((\ln n)^{-(s+1)/2})$.

A two-dimensional magnetic periodic Schrodinger operator with a variable metric is considered. An electric potential is assumed to be a distribution formally given by an expression \(\frac{{dv}}{{dx}}\), where dν is a periodic signed measure with a locally finite variation. We also assume that the perturbation generated by the electric potential is strongly subject (in the sense of forms) to the free operator. Under this natural assumption, we prove that the spectrum of the Schrodinger operator is absolutely continuous. Bibliography: 15 titles.

We consider a quasi-regular Dirichlet form. We show that a bounded signed measure charges no set of zero capacity associated with the form if and only if the measure can be decomposed into the sum of an integrable function and a bounded linear functional on the domain of the form. The decomposition allows one to describe explicitly the set of bounded measures charging no sets of zero capacity for interesting classes of Dirichlet forms. By way of illustration, some examples are given., arXiv admin note: text overlap with arXiv:1307.0717

W. Kleiner established (1964, Ann. Polon. Math.14, 117–130) for smooth curves and arcs an estimate for the discrepancy of a signed measure by using its energy norm. We extend this result to quasiconformal curves and arcs. The proof uses the theory of quasiconformal mappings and condenser theory. In a first step, the discrepancy of a signed measure can be estimated from above in terms of its energy norm and the capacity of a special condenser. This result is valid on every Jordan curve. Examples show the sharpness of the results from various points of view.

We consider L p -discrepancy estimates between two Borel measures supported on the interval [-1, 1] and give applications to the distribution of zeros of orthogonal polynomials. Thereby we obtain a new interpretation to the different local zero behavior of Pollaczek polynomials.

This article analyzes in detail the statistical and measure-theoretical properties of the nonuniform stationary measure, referred to as the w-invariant measure, associated with the spatial length distribution of the integral manifolds of the unstable invariant foliation in two-dimensional differentiable area-preserving systems. The analysis is developed starting from a sequence of analytical approximations for the associated density. These approximations are related to the properties of the Jacobian matrix of the $n\mathrm{th}$ iteration of a Poincar\'e map. The w-invariant measure plays a fundamental role in the study of transport phenomena in laminar-chaotic fluid-mixing systems, for which it furnishes the asymptotic invariant distribution of intermaterial contact length between two fluids. The w-invariant measure turns out to be singular and exhibits multifractal features. Its associated density displays local self-similarity in an \ensuremath{\varepsilon} neighborhood of hyperbolic periodic points. The cancellation exponent of the signed measure associated with the w measure by attaching at each point the direction of the field of the asymptotic unstable eigenvectors is also analyzed. The only case for which the w-invariant measure is absolutely continuous is given by the conjugation of hyperbolic toral automorphisms with a linear automorphism. The connections with the statistical properties, and in particular with the stretching dynamics, are addressed in detail.

Guided by analogy with Euler's spherical excess formula, we define a finite-additive functional on bounded convex polygons in ℝ2 (the Euler functional). Under certain smoothness assumptions, we find some sufficient conditions when this functional can be extended to a planar signed measure. A dual reformulation of these conditions leads to signed measures in the space of lines in ℝ2. In this way we obtain two sets of conditions which ensure that a segment function corresponds to a signed measure in the space of lines. The latter conditions are also necessary.

The historical significance of the development of measure theory is that it created a base for a generalization of the classical Riemann notion of the definite integral (which since 1854 has been considered to be the most general theory of integration). Riemann defined a bounded function over an interval [a, b] to be integrable if and only if the Darboux (or Cauchy) sums \(\sum^n_{i=1}f(t_i)\lambda(I_i)\) where \(\sum^n_{i=1}(I_i)\) is a finite decomposition of [a, b] into subintervals, approach a unique limiting value whenever the length of the largest subinterval goes to zero. The French mathematician, Henri Lebesgue (1875–1941), assumed that the above subintervals \((I_i)\) may be substituted by more general measurable sets and, in addition, that the class of Riemann integrable functions can be enlarged to the class of measurable functions. In this case, we arrive at a more advanced theory of integration, which is better suited for dealing with various limit processes and which led to the contemporary theory of probability and stochastic processes.

The present paper is devoted to signed measures of fuzzy quantum spaces. The Lebesque decomposition theorem is proved for the fuzzy case.

We prove dispersive estimates for the linear Schr\"odinger evolution associated to an operator -\Delta + V, where the potential is a signed measure of fractal dimension at least 3/2., Comment: 15 pages

The underlying mathematical structure of Shannon's information measures was studied in a paper by R.W. Yeung (1991), and the I-Measure mu *, which is a signed measure defined on a proper sigma -field F, was introduced. The I-Measure is a natural extension of Shannon's information measures and is uniquely defined by them. They also introduced as a consequence the I-Diagram as a geometric tool for visualizing the relationship among the information measures. In general, an I-Diagram for n random variables must be constructed in n-1 dimensions. It is shown that for any finite collection of random variables forming a Markov chain, mu * assumes a very simple structure which can be illustrated by an I-Diagram in two dimensions, and mu * is a nonnegative measure. >

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process $Y$ is a diffusion process whose generator can be formally written as $L+\mu\cdot\nabla-\nu$ with Dirichlet boundary conditions, where $L$ is a uniformly elliptic second-order differential operator and $\mu=(\mu^1,...,\mu^d)$ is such that each component $\mu^i$, $i=1,...,d$, is a signed measure belonging to the Kato class $\mathbf{K}_{d,1}$ and $\nu$ is a (nonnegative) measure belonging to the Kato class $\mathbf{K}_{d,2}$. We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for $Y$. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion $Y^D$ with measure-valued drift and potential when $D$ is one of the following types of bounded domains: twisted H\"{o}lder domains of order $\alpha\in(1/3,1]$, uniformly H\"{o}lder domains of order $\alpha\in(0,2)$ and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181--206] and [Probab. Theory Related Fields 91 (1992) 405--443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of $Y^D$ is finite., Comment: Published in at http://dx.doi.org/10.1214/07-AOP381 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider one-dimensional stochastic differential equations with generalized drift which involve the local time $L^X$ of the solution process: X_t = X_0 + \int_0^t b(X_s) dB_s + \int_\mathbb{R} L^X(t,y) \nu(dy), where b is a measurable real function, $B$ is a Wiener process and $\nu$ denotes a set function which is defined on the bounded Borel sets of the real line $\mathbb{R}$ such that it is a finite signed measure on $\mathscr{B}([-N,N])$ for every $N \in \mathbb{N}$. This kind of equation is, in dependence of using the right, the left or the symmetric local time, usually studied under the atom condition $\nu({x}) -1/2$ and $|\nu({x})| < 1$, respectively. This condition allows to reduce an equation with generalized drift to an equation without drift and to derive conditions on existence and uniqueness of solutions from results for equations without drift. The main aim of the present note is to treat the cases $\nu({x}) \geq 1/2$, $\nu({x}) \leq -1/2$ and $|\nu({x})| \geq 1$, respectively, for some $x \in \mathbb{R}$, and we give a complete description of the features of equations with generalized drift and their solutions in these cases.

Since its foundation by Borel and Lebesgue around the year 1900 the modern theory of measure, generalizing the basic notions of length, area and volume, has become one of the major fields in Pure and Applied Mathematics. In all human activities one collects measurements subject to variability and leading to the classical concepts of Probability and Mathematical Statistics that can modelize “observations” : random variables, samples, point processes. All of them belong to Measure Theory. The study of point processes and more generally of random measures has recently known a large development. It requires sophisticated mathematical tools. The very definition of random measures raises delicate problems, just as the need for a notion of closeness between them.

We consider the approximation of the distribution of the sum of independent but not necessarily identically distributed random variables by a compound Poisson distribution and also by a finite signed measure of higher accuracy. Using Kerstan's method, some new bounds for the total variation distance are presented. Recently, several authors had difficulties applying Stein's method to the problem given. For instance, Barbour, Chen and Loh used this method in the case of random variables on the nonnegative integers. Under additional assumptions, they obtained some bounds for the total variation distance containing an undesirable log term. In the present paper, we shall show that Kerstan's approach works without such restrictions and yields bounds without log terms.

The higher-order heat-type equation $ \partial u/\partial t=\pm\partial^{n} u/ \partial x^{n} $ has been investigated by many authors. With this equation is associated a pseudo-process $(X_t)_{t\ge 0}$ which is governed by a signed measure. In the even-order case, Krylov (1960) proved that the classical arc-sine law of Paul Levy for standard Brownian motion holds for the pseudo-process $(X_t)_{t\ge 0}$, that is, if $T_t$ is the sojourn time of $(X_t)_{t\ge 0}$ in the half line $(0,+\infty)$ up to time $t$, then $P(T_t\in ds)=\frac{ds}{\pi\sqrt{s(t-s)}}$, $0 \lt s \lt t$. Orsingher (1991) and next Hochberg and Orsingher (1994) obtained a counterpart to that law in the odd cases $n=3,5,7.$ Actually Hochberg and Orsingher (1994) proposed a more or less explicit expression for that new law in the odd-order general case and conjectured a quite simple formula for it. The distribution of $T_t$ subject to some conditioning has also been studied by Nikitin & Orsingher (2000) in the cases $n=3,4.$ In this paper, we prove that the conjecture of Hochberg and Orsingher (1994) is true and we extend the results of Nikitin & Orsingher for any integer $n$. We also investigate the distributions of maximal and minimal functionals of $(X_t)_{t\ge 0}$, as well as the distribution of the last time before becoming definitively negative up to time $t$.

Let $L_n$ be the length of the longest increasing subsequence of a random permutation of the numbers $1,\ldots, n$, for the uniform distribution on the set of permutations. Hammersley's interacting particle process, implicit in Hammersley (1972), has been used in Aldous and Diaconis (1995) to provide a “soft” hydrodynamical argument for proving that $\lim_{n \to \infty} EL_n / \sqrt{n} = 2$. We show in this note that the latter result is in fact an immediate consequence of properties of a random 2­dimensional signed measure, associated with Hammersley’s process.

$$ \left| {\smallint _{{\text{R}}^n } f\left( x \right)g\left( x \right)dx} \right| \leqslant C\left( n \right)\left\| {S_1 \left( {tD_t u} \right)} \right\|_{L^1 } \left\| g \right\|_{{\text{BMO,}}} $$ (12.1) where $$ f,g \in L^2 , $$ (12.2) $$ u\left( {x,t} \right) = \smallint _{{\text{R}}^n } P\left( {x - y,t} \right)f\left( y \right)dy. $$ (12.3) (In (12.3), P (x, t) denotes the Poisson kernel.) Since \( \left\| {S_1 \left( {tD_t u} \right)} \right\|_{L^1 } \) dominates \( \left\| f \right\|_{H^1 } \) by (10.19), (12.1) follows from Lemma 2.2 (with p = 1) and Theorem 2.2. In this section, we explain C. Fefferman’s direct proof of (12.1), which is one of the oldest proofs of his H 1-BMO duality theorem. (Another one of the oldest proofs will be explained in Section 19.)

For a quasi-regular (symmetric) Dirichlet space $( {\cal E}, {\cal F})$ and an associated symmetric standard process $(X_t, P_x)$, we show that, for $u in {\cal F}$, the additive functional $u^*(X_t) - u^*(X_0)$ is a semimartingale if and only if there exists an ${\cal E}$-nest $\{F_n\}$ and positive constants $C_n$ such that $ \vert {\cal E}(u,v)\vert \leq C_n \Vert v\Vert_\infty, v \in {\cal F}_{F_n,b}.$ In particular, a signed measure resulting from the inequality will be automatically smooth. One of the variants of this assertion is applied to the distorted Brownian motion on a closed subset of $R^d$, giving stochastic characterizations of BV functions and Caccioppoli sets.

Preprint enviat per a la seva publicació en una revista científica: The Annals of Statistics. Volume 25, Number 6 (1997), 2388-2409. [https://projecteuclid.org/euclid.aos/1030741078], A general way of constructing classes of goodness-of-fit tests for multivariate samples is presented. These tests are based on a random signed measure that plays the same role as the empirical process in the construction of the classical Kolmogorov-Smirnov tests. The resulting tests are consistent against any fixed alternative, and, for each sequence of contiguous alternatives, a test in each class can be chosen so as to optimize the discrimination of those alternatives.

Continuing the work of the preceding chapter on continuous parameter (sub)martingales, we present a solution of the Doob decomposition problem, raised in Section 2.5, which is due to Meyer. We give an elementary (but longer) demonstration and also sketch a shorter (but more sophisticated) argument based on Doleans-Dade signed measure representation of quasimartingales. This decomposition leads to stochastic integration with square integrable martingales as integrators generalizing the classical Ito integration. The material is presented in this chapter in considerable detail, since it forms a basis for semimartingale integrals with numerous applications to be abstracted and treated in the following chapter. Orthogonal decompositions of square integrable martingales (of continuous time parameter), its time change transformation leading to a related Brownian motion process and the Levy characterization of the latter from continuous parameter martingales, are covered. Stopping (or optional) times play a key role in all this work, and some classifications of these are given. The treatment also includes the Stratonovich integrals as well as an identification of the square integrable martingale integrators with spectral measures of certain normal operators in Hilbert space. Finally some related results appear as exercises in the Complements section.

In this paper we present a new way for proving the existence of non-measurable sets using a convenient operation of a discrete group on the Euclidian sphere. The only choice assumption used in this construction is the Hahn-Banach theorem, a weaker hypothesis than the Boolean Prime Ideal Theorem. Our construction proves that the Hahn-Banach theorem implies the existence of a non-Lebesgue-measurable set of reals. In fact we prove (under Hahn-Banach theorem) that there is no finitely additive, rotation invariant extension of Lebesgue measure to all subsets of the three-dimensional Euclidean space.

We study positive continuous additive functionals of diffusion processes associated with Dirichlet forms. We also give necessary and sufficient conditions for a Feynman-Kac semigroup associated with a signed Borel measure to be a strongly continuous semigroup in the relevant L 2-space. We characterize their generators as second order partial differential operators with zero order term (potential) given by a signed measure (which can be so singular as to be nowhere Radon). We also discuss preservation of p-boundedness and L p -smoothing properties of semigroups under perturbations. We also study integral kernels of Feynman-Kac semigroups and provide upper bounds for them.

Perturbation by differences of unbounded potentials and corresponding eigenvalues to the time-independent Schrodinger equation yields the following: Let μ be a signed measure on R n such that μ does not charge polar sets and μ − generates a finite potential. Let U be a bounded domain in R n such that U is (μ+βλ n )-bounded for some β≥0 (λ n Lebesgue measure). Then the set of all eigenvalues of the equation (1/2Δ-μ)u=0 on U is a non-empty upper bounded subset of R which has no accumulation points

Selberg has shown on the basis of the Riemann hypothesis that for every ε > 0 most intervals |x,x+xϵ| of length xϵ contain approximately xϵlogx primes. Here by “most” we mean “for a set of values of x of asymptotic density one.” Prachar has extended Selberg's result to primes in arithmetic progressions. Both authors noted that if we assume the quasi Riemann hypothesis, that ζ(s) has no zeros in the domain {σ>12+δ} for some δ 2 δ. Here we give a simple proof of these theorems in a general context, where an arbitrary signed measure takes the place of d[ψ(x)−x]. Then we show by a counterexample that this general theorem is the best of its kind: the condition ε > 2δ cannot be replaced by ε = 2δ. In our example, the associated Dirichlet integral is an entire function which remains bounded on the domain {σ≥12+δ}. Thus its growth and regularity properties are better than those of ζ′(s)ζ(s). Nevertheless the corresponding signed measure behaves badly.

We consider the equation $dX(t) = (\int_{ - r}^0 {X(s + t)d\mu (s))dt} + GdW(t)$ where $r > 0$, $\mu $ is a matrix-valued signed measure on $[ - r,0]$, W is an n-dimensional Wiener process and G is a fixed $n \times n$ matrix. Using the known decomposition $C = S \oplus U$ of $C = C[ - r,0]$ into a stable closed subspace S and a finite dimensional unstable subspace U for the analogous deterministic equation, we prove that the projection of the solution X onto S converges exponentially fast to a stationary Gaussian process. Also the speed of explosion of the projection onto U is exhibited.

Let μ be a signed measure with a density. If the components of p: R n → R d are linearly independent polynomials, an estimate for the Fourier transform of the signed measure μp−1 on R d is derived.

Let u = u ( x , t ) u = u(x,t) be a continuous weak solution of the porous medium equation in R d × ( 0 , T ) {{\mathbf {R}}^d} \times (0,T) for some T > 0 T > 0 . We show that corresponding to u u there is a unique nonnegative Borel measure ρ \rho on R d {{\mathbf {R}}^d} which is the initial trace of u u . Moreover, we show that the initial trace ρ \rho must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then | x | 2 |x{|^2} as | x | → ∞ |x| \to \infty .

We study the Hamiltonians for nonrelativistic quantum mechanics in one dimension, in terms of energy forms ∫‖df/dx‖2 dx+∫‖ f ‖2 d( μ −ν), where μ and ν are positive, not necessarily finite measures on the real line. We cover, besides regular potentials, cases of very singular interactions (e.g., a particle interacting with an infinite number of fixed particles by ‘‘delta function potentials’’ of arbitrary strengths). We give conditions for lower semiboundedness and closability of the above energy forms, which are sufficient and, for certain classes of potentials (e.g., μ−ν a signed measure), also necessary. In contrast to the results in other approaches, no regularity conditions and no restrictions on the growth of the measures μ and ν at infinity are needed.