Related papers: Explicit representations for multiscale L\'evy pro…
We study the relaxation process in normal and anomalous diffusion regimes for systems described by a generalized Langevin equation (GLE). We demonstrate the existence of a very general correlation function which describes the relaxation…
Using complex analysis techniques we obtain precise asymptotic approximations for the kernels corresponding to the symmetric $\alpha$-stable processes and their fractional derivatives. We apply our method to general L\'evy processes whose…
The paper investigates nonlinear control laws obtained from linear one by two types of substitutions using odd functions. The first substitution consists in passing each component of the state vector through a nonlinear function, the second…
In this paper a differential equation with noninteger order was used to model an anomalous luminescence decay process. Although this process is in principle an exponential decaying process, recent data indicates that is not the case for…
Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of…
Nonlinear scalar conservation laws are traditionally viewed as transport equations. We take instead the viewpoint of these PDEs as continuity equations with an implicitly defined velocity field. We show that a weak solution is the entropy…
We study the relation between L\'evy processes under nonlinear expectations, nonlinear semigroups and fully nonlinear PDEs. First, we establish a one-to-one relation between nonlinear L\'evy processes and nonlinear Markovian convolution…
The conservation laws for a class of nonlinear equations with variable coefficients on discrete and noncommutative spaces are derived. For discrete models the conserved charges are constructed explicitly. The applications of the general…
We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for…
The paper is concerned with a scalar conservation law with discontinuous gradient-dependent flux. Namely, the flux is described by two different functions $f(u)$ or $g(u)$, when the gradient $u_x$ of the solution is positive or negative,…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…
Recent advances have allowed to tackle exact path-space probabilistic representations of macroscopic advection-diffusion models involving advection nonlinearities by step forward approaches in terms of continuous branching stochastic…
In this paper, we study the asymptotic behavior for multi-scale stochastic differential equations driven by L\'evy processes. The optimal strong convergence order 1/2 is obtained by studying the regularity estimates for the solution of…
We introduce a Langevin equation characterized by a time dependent drift. By assuming a temporal power-law dependence of the drift we show that a great variety of behavior is observed in the dynamics of the variance of the process. In…
In this paper, we develop a numerical method for the L\'evy-Fokker-Planck equation with the fractional diffusive scaling. There are two main challenges. One comes from a two-fold nonlocality, that is, the need to apply the fractional…
As an analogue to the explicit formula in the stable case, the asymptotic behavior at the origin of the renormalized zero resolvent of one-dimensional L\'evy processes is studied under certain regular variation conditions on the…
The non-Markovian continuous-time random walk model, featuring fat-tailed waiting times and narrow distributed displacements with a non-zero mean, is a well studied model for anomalous diffusion. Using an analytical approach, we recently…
Anomalous dynamics in which local perturbations spread faster than diffusion are ubiquitously observed in the long-time behavior of a wide variety of systems. Here, the manner by which such systems evolve towards their asymptotic…
We study the relation between flow structure and fluid deformation in steady two-dimensional random flows. Beyond the linear (shear flow) and exponential (chaotic flow) elongation paradigms, we find a broad spectrum of stretching behaviors,…
In the present work, we establish the approximation of nonlinear stochastic partial differential equation (SPDE) driven by cylindrical {\alpha}-stable L\'evy processes via modulation or amplitude equations. We study SPDEs with a cubic…