Anomalous Transport and Anomalous Diffusion

Scientific Programme

Lectures

• Luisa Beghin - Anomalous Diffusion, Lévy Processes, and Fractional Calculus

  The aim of the course is to provide a comprehensive introduction to the mathematical foundations of anomalous diffusion and its modeling through fractional calculus.

  As a first step, we will review Markov processes and introduce Lévy processes as stochastic models characterized by stationary and independent increments. We will then explore the fundamental concepts of fractional operators (e.g. Riemann-Liouville, Caputo, Riesz derivatives and integrals), highlighting their inherent non-locality and application in modeling physical memory effects. We also analyze the connection between the microscopic dynamics—specifically the Continuous-Time Random Walk (CTRW) model—and its macroscopic limit via the Stable Central Limit Theorem, which generates the stable Lévy distributions. Finally, the course will cover the derivation and solution of fractional diffusion equations both in the time- and the space-fractional cases, as tools for modeling sub- and super-diffusions.

• Zhen-Qing Chen and Takashi Kumagai - Anomalous subdiffusion and time-fractional differential equations

  Anomalous diffusion phenomenon has been observed in many natural systems, from the signaling of biological cells, to the foraging behavior of animals, to the travel times of contaminants in groundwater. This short course will consist of two parts.

  The first part will discuss the interplay between anomalous sub-diffusions and time-fractional differential equations, including how they arise naturally from limit theorems for random walks. We will then present some recent results in this area, in particular on the probabilistic representation to the solutions of time fractional equations with source terms. An interesting feature of the latter is that they involve two fundamental solutions. The second part will discuss applications of anomalous sub-diffusions to scaling limits of trap models, and the two-sided estimates of the fundamental solutions arise in the time-fractional parabolic equations.

• Tomasz Komorowski and Stefano Olla - Heat superdiffusion in one-dimensional oscillators chains

  This course provides an in-depth study of anomalous diffusion and energy transport in classical models of statistical mechanics, with particular emphasis on one-dimensional chains of interacting oscillators. Such systems originate in Debye’s early twentieth-century model of energy propagation in solids and achieved renewed significance through the seminal Fermi–Pasta–Ulam–Tsingou numerical experiments of the 1950s, which investigated the mechanisms by which oscillator chains approach thermal equilibrium. One-dimensional systems are especially compelling, as they generically exhibit anomalous—rather than classical diffusive—energy transport.
  After introducing the physical motivation and historical context, the course develops the hydrodynamic limits associated with conserved quantities in stochastically perturbed oscillator chains. Under hyperbolic scaling, in which space and time are rescaled at the same rate, anharmonic chains with stochastic perturbations give rise to macroscopic Euler-type equations. On longer time scales, the focus shifts to harmonic chains. In the presence of momentum-conserving stochastic perturbations, energy transport becomes superdiffusive and is governed by a heat equation involving a fractional Laplacian.
  The course concludes with an analysis of kinetic limits for both closed systems, which evolve in isolation, and open systems interacting with external heat reservoirs or external forcing. Particular attention is paid to the emergence of macroscopic boundary conditions and their role in the resulting transport equations.

Seminars

• Giada Basile - TBA

  TBA

• Krzysztof Bogdan - Stable processes with reflections

  Markov processes underly evolutionary phenomena such as the transfer of heat or spreading of a population in a given area. Such movements can occur continuously or through jumps. Their statistical effects are described by solving parabolic and elliptic equations. Dirichlet conditions in such equations are related to absorption of particles. They are well studied, because the stopped process is easy to define. The more difficult and varied are Neumann conditions, associated with reflection of particles. In particular, the censored process [1] is an instance of Neumann-type conditions. But there are more and they are challenging, because each boundary condition actually requires construction of the corresponding transition semigroup [4] and process [5]. The topic is of much interest for PDEs [2, 3, 6], too. We will mainly discuss specific jump Markov processes and non-local boundary Neumann conditions from [5, 6] and some work in progress.

  [1] K. Bogdan, K. Burdzy, Z.-Q. Chen, Censored stable processes, Probab. Theory Related Fields, 2003, 127(1), 89–152.
  [2] S. Dipierro, X. Ros-Oton, E. Valdinoci, Nonlocal problems with Neumann boundary conditions, Rev. Mat. Iberoam., 2017, 33(2), 377–416.
  [3] K. Bogdan, T. Grzywny, K. Pietruska-Pałuuba, A. Rutkowski, Extension and trace for nonlocal operators, J. Math. Pures Appl. (9), 2020, 137, 33–69.
  [4] K. Bogdan, M. Kunze, The fractional Laplacian with reflections, Potential Anal., 2024, 61(2), 317–345.
  [5] K. Bogdan, M. Kunze, Stable processes with reflections, Stochastic Process. Appl., 2025, 187, Paper No. 104654.
  [6] K. Bogdan, D. Fafuła, P. Sztonyk, Nonlocal operators with Neumann conditions, Dissertationes Math., 2025, 604, 91 pp.

• Tomasz Grzywny - Stationary states for Lévy processes with resetting

  In this talk we consider a $d$-dimensional stochastic process $X$ which arises from a L\'evy process $Y$ by partial resetting, that is, the position of the process $X$ at a Poisson moment equals $c \in (0,1)$ times its position right before the moment, and it develops as $Y$ between these two consecutive moments.
  We focus on $Y$ being a strictly $\alpha$-stable process with $\alpha\in (0,2]$ having a transition density. We analyze properties of the transition density $p$ of the process $X$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $\rho_{Y}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $Y$ starting from zero, which results in closed concise formulae for its moments. We show that the process $X$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker-Planck equation, and we confirm the harmonicity of $\rho_{Y}$ with respect to the adjoint generator.
  In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $\alpha$-stable processes for $\alpha \in (0,2)$, and $\alpha$-stable subordinator for $\alpha\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is, a change of the asymptotic behavior of $p(t;0,y)$ with respect to $\rho_{Y}(y)$.

• Jozsef Lorinczi - TBA

  TBA

• Michal Ryznar - TBA

  TBA

• Marielle Simon - Standard and anomalous diffusion of energy in chains of coupled oscillators

  Over the last few years, anomalous behaviors have been observed for one-dimensional chains of oscillators. The rigorous derivation of such behaviors from deterministic systems of Newtonian particles is very challenging, due to the existence of conservation laws, which impose very poor ergodic properties to the dynamical system. A possible way out of this lack of ergodicity is to introduce stochastic models, in such a way that the qualitative behaviour of the system is not modified. One starts with a chain of oscillators with a Hamiltonian dynamics, and then adds a stochastic which keeps the fundamental conservation laws (energy, momentum and stretch, usually).
  The following result is now well-known: when the one-dimensional system is given by an unpinned harmonic chain of oscillators perturbed by an energy-momentum conserving noise, the energy fluctuation field at equilibrium evolves according to an infinite dimensional 3/4-fractional Ornstein-Uhlenbeck process. On the other hand, when the velocities of particles can randomly change sign (and therefore the only conserved quantities of the dynamics are the energy and the stretch), under a diffusive space-time scaling, the energy profile evolves following a non-linear diffusive equation involving the stretch.
  In this talk I will consider the same harmonic Hamiltonian dynamics, now perturbed by both stochastic noises, and I will describe the regime transition for the energy fluctuations. We will see that the limit of the energy fluctuation field depends on the evanescent speed of the random perturbation, that we can recover the two very different regimes for the energy transport, and additionally we exhibit a new macroscopic operator at the critical value of the intensity. This talk is based on a collaborative work with C. Bernardin, P. Gonçalves, M. Jara and M. Sasada.

• Bruno Toaldo - TBA

  TBA

• Vanja Wagner - TBA

  TBA

• Cedric Bernardin - TBA

  TBA