Reflected Backward Stochastic Differential Equation with Jumps and RCLL Obstacle

In this paper we study one-dimensional reflected backward stochastic differential equation when the noise is driven by a Brownian motion and an independent Poisson point process when the solution is forced to stay above a right continuous left-hand limited obstacle. We prove existence and uniqueness of the solution by using a penalization method combined with a monotonic limit theorem.

Hopf bifurcation of a delay differential equation with two delays

We consider a delay differential equation with two delays. The Hopf bifurcation of this equation is investigated together with the stability of the bifurcated periodic solution, its period and the bifurcation direction. Finally, three applications are given.

Rigorous derivation of a nonlinear diffusion equation as fast-reaction limit of a continuous coagulation-fragmentation model with diffusion

Weak solutions of the spatially inhomogeneous (diffusive) Aizenmann-Bak model of coagulation-breakup within a bounded domain with homogeneous Neumann boundary conditions are shown to converge, in the fast reaction limit, towards local equilibria determined by their mass. Moreover, this mass is the solution of a nonlinear diffusion equation whose nonlinearity depends on the (size-dependent) diffusion coefficient. Initial data are assumed to have integrable zero order moment and square integrable first order moment in size, and finite entropy. In contrast to our previous result [CDF2], we are able to show the convergence without assuming uniform bounds from above and below on the number density of clusters.

On the analysis of travelling waves to a nonlinear flux limited reaction-diffusion equation

In this paper we study the existence and qualitative properties of travelling waves associated to a nonlinear flux limited partial differential equation coupled to a Fisher-Kolmogorov-Petrovskii-Piskunov type reaction term. We prove the existence and uniqueness of finite speed moving fronts of C2 classical regularity, but also the existence of discontinuous entropy travelling wave solutions.

Wellposedness of a nonlinear, logarithmic Schrödinger equation of Doebner-Goldin type modeling quantum dissipation

This paper is concerned with the modeling and analysis of quantum dissipation phenomena in the Schrödinger picture. More precisely, we do investigate in detail a dissipative, nonlinear Schrödinger equation somehow accounting for quantum Fokker–Planck effects, and how it is drastically reduced to a simpler logarithmic equation via a nonlinear gauge transformation in such a way that the physics underlying both problems keeps unaltered. From a mathematical viewpoint, this allows for a more achievable analysis regarding the local wellposedness of the initial–boundary value problem. This simplification requires the performance of the polar (modulus–argument) decomposition of the wavefunction, which is rigorously attained (for the first time to the best of our knowledge) under quite reasonable assumptions.

Symmetries and fixed point stability of stochastic differential equations modeling self-organized criticality

A stochastic nonlinear partial differential equation is constructed for two different models exhibiting self-organized criticality: the Bak-Tang-Wiesenfeld (BTW) sandpile model [Phys. Rev. Lett. 59, 381 (1987); Phys. Rev. A 38, 364 (1988)] and the Zhang model [Phys. Rev. Lett. 63, 470 (1989)]. The dynamic renormalization group (DRG) enables one to compute the critical exponents. However, the nontrivial stable fixed point of the DRG transformation is unreachable for the original parameters of the models. We introduce an alternative regularization of the step function involved in the threshold condition, which breaks the symmetry of the BTW model. Although the symmetry properties of the two models are different, it is shown that they both belong to the same universality class. In this case the DRG procedure leads to a symmetric behavior for both models, restoring the broken symmetry, and makes accessible the nontrivial fixed point. This technique could also be applied to other problems with threshold dynamics.

Semigroup analysis of structured populations with distributed states-at-birth arising in mathematical aquaculture

Motivated by the modelling of structured parasite populations in aquaculture we consider a class of physiologically structured population models, where individuals may be recruited into the population at different sizes in general. That is, we consider a size-structured population model with distributed states-at-birth. The mathematical model which describes the evolution of such a population is a first order nonlinear partial integro-differential equation of hyperbolic type. First, we use positive perturbation arguments and utilise results from the spectral theory of semigroups to establish conditions for the existence of a positive equilibrium solution of our model. Then we formulate conditions that guarantee that the linearised system is governed by a positive quasicontraction semigroup on the biologically relevant state space. We also show that the governing linear semigroup is eventually compact, hence growth properties of the semigroup are determined by the spectrum of its generator. In case of a separable fertility function we deduce a characteristic equation and investigate the stability of equilibrium solutions in the general case using positive perturbation arguments.

Unified formalism for non-autonomous mechanical systems

We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of regular and non-regular time-dependent mechanical systems, which is based on the approach of Skinner and Rusk (1983). The dynamical equations of motion and their compatibility and consistency are carefully studied, making clear that all the characteristics of the Lagrangian and the Hamiltonian formalisms are recovered in this formulation. As an example, it is studied a semidiscretization of the nonlinear wave equation proving the applicability of the proposed formalism.

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results.

Stochastic differential equations with random coefficients

In this paper we establish the existence and uniqueness of a solution for different types of stochastic differential equation with random initial conditions and random coefficients. The stochastic integral is interpreted as a generalized Stratonovich integral, and the techniques used to derive these results are mainly based on the path properties of the Brownian motion, and the definition of the Stratonovich integral.

Stochastic delay differential equations driven by fractional Brownian motion with Hurst parameter H 1/2

We consider the Cauchy problem for a stochastic delay differential equation driven by a fractional Brownian motion with Hurst parameter H>¿. We prove an existence and uniqueness result for this problem, when the coefficients are sufficiently regular. Furthermore, if the diffusion coefficient is bounded away from zero and the coefficients are smooth functions with bounded derivatives of all orders, we prove that the law of the solution admits a smooth density with respect to Lebesgue measure on R.

Weierstrass integrability of differential equations

The integrability problem consists in finding the class of functions a first integral of a given planar polynomial differential system must belong to. We recall the characterization of systems which admit an elementary or Liouvillian first integral. We define {\it Weierstrass integrability} and we determine which Weierstrass integrable systems are Liouvillian integrable. Inside this new class of integrable systems there are non--Liouvillian integrable systems.

Large deviation for BSDE with subdifferential operator

In this paper we prove that the solution of a backward stochastic differential equation, which involves a subdifferential operator and associated to a family of reflecting diffusion processes, converges to the solution of a deterministic backward equation and satisfes a large deviation principle.

Well-posedness and invariant measures for HJM models with deterministic volatility and Lévy noise

We give sufficient conditions for existence, uniqueness and ergodicity of invariant measures for Musiela's stochastic partial differential equation with deterministic volatility and a Hilbert space valued driving Lévy noise. Conditions for the absence of arbitrage and for the existence of mild solutions are also discussed.

Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.