Space Of Radon Measures Research Articles

A fitness-driven cross-diffusion system from population dynamics as a gradient flow

We consider a fitness-driven model of dispersal of N interacting populations, which was previously studied merely in the case N=1. Based on some optimal transport distance recently introduced, we identify the model as a gradient flow in the metric space of Radon measures. We prove existence of global non-negative weak solutions to the corresponding system of parabolic PDEs, which involves degenerate cross-diffusion. Under some additional hypotheses and using a new multicomponent Poincaré–Beckner functional inequality, we show that the solutions converge exponentially to an ideal free distribution in the long time regime.

Mass concentration in a nonlocal model of clonal selection.

Self-renewal is a constitutive property of stem cells. Testing the cancer stem cell hypothesis requires investigation of the impact of self-renewal on cancer expansion. To better understand this impact, we propose a mathematical model describing the dynamics of a continuum of cell clones structured by the self-renewal potential. The model is an extension of the finite multi-compartment models of interactions between normal and cancer cells in acute leukemias. It takes a form of a system of integro-differential equations with a nonlinear and nonlocal coupling which describes regulatory feedback loops of cell proliferation and differentiation. We show that this coupling leads to mass concentration in points corresponding to the maxima of the self-renewal potential and the solutions of the model tend asymptotically to Dirac measures multiplied by positive constants. Furthermore, using a Lyapunov function constructed for the finite dimensional counterpart of the model, we prove that the total mass of the solution converges to a globally stable equilibrium. Additionally, we show stability of the model in the space of positive Radon measures equipped with the flat metric (bounded Lipschitz distance). Analytical results are illustrated by numerical simulations.

On a pseudoparabolic regularization of a forward–backward–forward equation

We consider an initial–boundary value problem for a degenerate pseudoparabolic regularization of a nonlinear forward–backward–forward parabolic equation, with a bounded nonlinearity which is increasing at infinity. We prove existence of suitably defined nonnegative solutions of the problem in a space of Radon measures. Solutions satisfy several monotonicity and regularization properties; in particular, their singular part is nonincreasing and may disappear in finite time. The problem is of intrinsic mathematical interest, but also arises naturally when studying, by time reversal, the spontaneous appearance of singularities in a specific application.

Asymptotic of Sparse Support Recovery for Positive Measures

We study sparse spikes deconvolution over the space of Radon measures when the input measure is a finite sum of positive Dirac masses using the BLASSO convex program. We focus on the recovery properties of the support and the amplitudes of the initial measure in the presence of noise when the minimum separation distance t of the input measure (the minimum distance between two spikes) tends to zero. We show that when ||ω||2/λ, ||ω||2/t2N-1 and λ/t2N-1 are small enough (where λ is the regularization parameter, ω the noise and N the number of spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a noise level and a regularization parameter small enough with respect to the minimum separation distance, there exists a unique solution to the BLASSO program with exactly the same number of spikes as the original measure. We provide an upper bound on the error with respect to the initial measure. As a by-product, we show that the amplitudes and positions of the spikes of the solution both converge towards those of the input measure when λ and ω drop to zero faster than t2N-1.

A moment problem for random discrete measures

Let X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η=∑isiδxi, where, for each i, si>0 and δxi is the Dirac measure at xi∈X. A random discrete measure on X is a probability measure on K(X). The main result of the paper states a necessary and sufficient condition (conditional upon a mild a priori bound) when a random measure μ is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure μ. Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterization via moments is given when a random measure is a point process.

Uniform equicontinuity, multiplier topology and continuity of convolution

We characterise bounded uniformly equicontinuous sets of functions on locally compact groups in terms of uniform factorisation. We apply this result to study the continuity of the convolution product on the dual LUC(G)* of the space of bounded left uniformly continuous functions with the topology of uniform convergence on bounded uniformly equicontinuous sets. When restricted to the space of finite Radon measures on a locally compact group, this is the right multiplier topology. For any topological group, the convolution is jointly continuous on bounded sets in the measure algebra. It is jointly continuous on all of LUC(G)* when G is a locally compact SIN group.

Conservation of ultraspherical convolution

The space of pseudomeasures \(\mathcal FL^\infty \) lies between the space of Radon measures and distributions; \( \mathcal M\subset \mathcal FL^\infty \subset (C_c^\infty )'\). The theory of strongly continuous semigroups is generalized by Lions (Port Math 19:141–164, 1960) to distribution semigroups (DSGs) \(\fancyscript{G}\in \fancyscript{L}(C_c^\infty ,\fancyscript{L}(X))\), where \(\fancyscript{L}(X)\) is the algebra of linear continuous operators on a Banach space \(X\), satisfying the conservation of convolution \(\fancyscript{G}(\varphi *\psi )=\fancyscript{G}(\varphi )\fancyscript{G}(\psi )\), for all \(\varphi , \psi \in C_c^\infty (\mathbb R)_+.\) In this paper we define the subspace \(\Pi _\nu \) of \(\fancyscript{L}(L^2([-1,1],\mu _\nu ))\) of pseudomeasure operators \(T\), satisfying \(T(g *_\nu h) = (Tg)*_\nu h, \) for any \( g, h \in L^2(I,\mu _\nu )\). We prove that \(T^2\) conserves the generalized convolution \(T^2(f*_\nu g)=T(f)*_\nu T(g)\) and study certain properties of DSGs with different types of endomorphism \(\fancyscript{G}=T^2\).

Splitting-particle methods for structured population models: Convergence and applications

We propose a new numerical scheme designed for a wide class of structured population models based on the idea of operator splitting and particle approximations. This scheme is related to the Escalator Boxcar Train (EBT) method commonly used in biology, which is in essence an analogue of particle methods used in physics. Our method exploits the split-up technique, thanks to which the transport step and the nonlocal integral terms in the equation can be separately considered. The order of convergence of the proposed method is obtained in the natural space of finite non-negative Radon measures equipped with the flat metric. This convergence is studied even adding reconstruction and approximation steps in the particle simulation to keep the number of approximation particles under control. We validate our scheme in several test cases showing the theoretical convergence error. Finally, we use the scheme in situations in which the EBT method does not apply showing the flexibility of this new method to cope with the different terms in general structured population models.

The weak McShane integral

We present a weaker version of the Fremlin generalized McShane integral (1995) for functions defined on a σ-finite outer regular quasi Radon measure space (S,Σ, T, µ) into a Banach space X and study its relation with the Pettis integral. In accordance with this new method of integration, the resulting integral can be expressed as a limit of McShane sums with respect to the weak topology. It is shown that a function f from S into X is weakly McShane integrable on each measurable subset of S if and only if it is Pettis and weakly McShane integrable on S. On the other hand, we prove that if an X-valued function is weakly McShane integrable on S, then it is Pettis integrable on each member of an increasing sequence (S l ) l⩾1 of measurable sets of finite measure with union S. For weakly sequentially complete spaces or for spaces that do not contain a copy of c 0, a weakly McShane integrable function on S is always Pettis integrable. A class of functions that are weakly McShane integrable on S but not Pettis integrable is included.

Pseudo-parabolic regularization of forward-backward parabolic equations: Power-type nonlinearities

Abstract We study a quasilinear parabolic equation of forward-backward type, under assumptions on the nonlinearity which hold for a wide class of mathematical models, using a pseudo-parabolic regularization of power type. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. It is shown that these solutions satisfy suitable entropy inequalities. We also study their qualitative properties, in particular proving that the singular part of the solution with respect to the Lebesgue measure is constant in time.

The strict topology for the space of Radon measures on a locally compact Hausdorff space

In this paper, we deal with the dual of the measure space M(X) for a locally compact Hausdorff space X under certain locally convex topologies. We first introduce and study a strict topology β(X) on M(X) under which its strong dual can be identified with L0∞(M(X)), the Banach space of all F∈L∞(M(X)) that vanishes at infinity. Under this duality, we investigate some attributes of (M(X),β(X)) as a locally convex space.

Best Minimizing Algorithm For Shape-measure Method

The Shape-Measure method for solving optimal shape design problems (OSD) in cartesian coordinates is divided into two steps. First, for a fixed shape (domain), the problem is transferred to the space of positive Radon measures and relaxed to a linear programming in which its optimal coefficients determine the optimal pair of trajectory and control. Then, a standard minimizing algorithm is used to identify the best shape. Here we deal with the best standard algorithm to identify the optimal solution for an OSD sample problem governed by an elliptic boundary control problem.

Some remarks on planar Boussinesq equations

The main purpose of this paper is to prove the well-posedness of the two-dimensional Boussinesq equations when the initial vorticity omega (0) a L (1)(R (2)) (or the finite Radon measure space). Using the stream function form of the equations and the Schauder fixed-point theorem to get the new proof of these results, we get that when the initial vorticity is smooth, there exists a unique classical solutions for the Cauchy problem of the two dimensional Boussinesq equations.

Inverse problems in spaces of measures

The ill-posed problem of solving linear equations in the space of vector-valued finite Radon measures with Hilbert space data is considered. Approximate solutions are obtained by minimizing the Tikhonov functional with a total variation penalty. The well-posedness of this regularization method and further regularization properties are mentioned. Furthermore, a flexible numerical minimization algorithm is proposed which converges subsequentially in the weak* sense and with rate O(n −1 )i n terms of the functional values. Finally, numerical results for sparse deconvolution demonstrate the applicability for a finite-dimensional discrete data space and infinite-dimensional solution space.

An age-structured two-sex model in the space of radon measures: Well posedness

In the following paper a well-posedness of an age-structured two-sex population model in a space of Radon measures equipped with a flat metric is presented. Existence and uniqueness of measure valued solutions is proved by a regularization technique. This approach allows to obtain Lipschitz continuity of solutions with respect to time and stability estimates. Moreover, a brief discussion on a marriage function, which is the main source of a nonlinearity, is carried out and an example of the marriage function fitting into this framework is given.

Degenerate Regularization of Forward–Backward Parabolic Equations: The Regularized Problem

We study a quasilinear parabolic equation of forward–backward type in one space dimension, under assumptions on the nonlinearity which hold for a number of important mathematical models (for example, the one-dimensional Perona–Malik equation), using a degenerate pseudoparabolic regularization proposed in Barenblatt et al. (SIAM J Math Anal 24:1414–1439, 1993), which takes time delay effects into account. We prove existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures. We also study qualitative properties of such solutions, in particular concerning their decomposition into an absolutely continuous part and a singular part with respect to the Lebesgue measure. In this respect, the existence of a family of viscous entropy inequalities plays an important role.

Some recent results concerning a class of forward-backward parabolic equations

We study a quasilinear parabolic equation of forward-backward type in one space dimension, under assumptions on the nonlinearity satisfied by important mathematical models (e.g., the one-dimensional Perona–Malik equation). We first address a degenerate pseudoparabolic regularization of the equation, which takes time delay e¤ects into account, proving existence and uniqueness of positive solutions of the regularized problem in a space of Radon measures, as well as qualitative properties of such solutions. Then we address the vanishing viscosity limit of the regularized problem, proving that the limiting points (in a suitable topology) of the family of solutions of the regularized problem can be regarded as solutions of the original problem. In particular, we characterize the disintegration of the narrow limit of the family of Young measures associated with the absolutely continuous part of the solutions of the regularized problem, proving that it is a superposition of two Dirac masses with support on the branches of the graph of the nonlinearity φ . In the above study, the existence of a family of infinitely many entropy inequalities plays an important role.

A Measure Theoretical Approach for Path Planning Problem of Nonlinear Control Systems

This paper presents a new approach to find an approximate solution for the nonlinear path planning problem. In this approach, first by defining a new formulation in the calculus of variations, an optimal control problem, equivalent to the original problem, is obtained. Then, a metamorphosis is performed in the space of problem by defining an injection from the set of admissible trajectory-control pairs in this space into the space of positive Radon measures. Using properties of Radon measures, the problem is changed to a measure-theo- retical optimization problem. This problem is an infinite dimensional linear programming (LP), which is approximated by a finite dimensional LP. The solution of this LP is used to construct an approximate solution for the original path planning problem. Finally, a numerical example is included to verify the effectiveness of the proposed approach.

A variational approach to norm attainment of some operators and polynomials

If X is an Asplund space, then every uniformly continuous function on BX* which is holomorphic on the open unit ball, can be perturbed by a w* continuous and homogeneous polynomial on X* to obtain a norm attaining function on the dual unit ball. This is a consequence of a version of Bourgain-Stegall’s variational principle. We also show that the set of N-homogeneous polynomials between two Banach spaces X and Y whose transposes attain their norms is dense in the corresponding space of N-homogeneous polynomials. In the case when Y is the space of Radon measures on a compact K, this result can be strengthened.

Measure-Valued Solutions for a Differential Game Related to Fish Harvesting

In this paper we consider a model for the harvesting of marine resources, described by an elliptic equation. Since the cost functionals have sublinear growth with respect to the pointwise intensity of fishing effort, optimal solutions are in general measure-valued. For the control problem in one space dimension, we prove the existence of optimal strategies. Uniqueness is established within a class of measures with small total mass. We also study the differential game, modeling the presence of several competing fishing companies, and prove the existence of a Nash equilibrium solution. This is obtained as a fixed point of a continuous transformation in a space of positive Radon measures.