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 - Large deviations and dynamical phase transitions

  I will present some results on large deviations in stochastic binary collision models. I will show that, although the dynamics conserves energy, atypical paths that violate energy conservation can occur with exponentially small probability as the number of particles increases. As a consequence, a dynamical phase transition may arise in the asymptotic behavior of the total number of collisions. Specifically, I will discuss examples related to energy transport and the role of anomalous diffusion in these processes.

• Cedric Bernardin - Fractional Macroscopic Fluctuation Theory

  Macroscopic Fluctuations Theory is the cornerstone of modern non-equilibrium statistical mechanics. Developed for boundary driven diffusive systems during the last 25 years, it has only been developed for interacting particle systems with short range interactions. In this talk we will propose to generalise this theory for microscopic models with long range interactions which are macroscopically described by a fractional diffusion equation.

• 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 - The location of extrema of solutions to a class of non-local equations in bounded domains

  I will consider a class of non-local heat equations in convex bounded domains under Dirichlet exterior condition, in which the kinetic operator is given by a Bernstein function of the Laplacian and the heat source/dissipation term is a multiplication operator, and derive a lower bound on the location of global extrema of their solutions from the boundary of the domain. Using that the kinetic operators are Markov generators of subordinate Brownian motion, the main tool of this analysis will be a stochastic representation of the solutions by running suitable jump processes and averaging over their paths. As it will be seen, this bound has an impact on various spectral geometric and variational quantities in terms of the survival times of the process.

• Michal Ryznar - On solutions of SDEs driven by singular Lévy noise

  In the talk we will study the following Stochastic Differential Equation
  $$ d X_t = b(X_t) \, dt + A(X_{t-}) \, d Z_t, \quad X_{0} = x \in \mathbb{R}^d, \quad t \ge 0 $$ where $A(x)$ is a family of square $d$-dimensional matrices and $Z=(Z^1, \dots, Z^d)^T$, with $Z^i, i=1,\dots, d$ being independent one-dimensional symmetric jump Lévy processes, not necessarily identically distributed. In particular, we partially cover the case when each $Z^i$ is one-dimensional symmetric $\alpha_i$-stable process ($\alpha_i \in (0,2)$ and they are not necessarily equal). Such noise is essentially singular, namely, the Lévy measure $\mu$ of $Z$ is concentrated on the collection of coordinate axes in $\mathbb{R}^d$, which combined with a non-trivial rotation occurring due to the matrix coefficient $A(x)$ yield lack of a single reference measure for the images of $\mu$ under the multiplicative mappings $z\mapsto A(x)z$ for different $x$. Under certain regularity assumptions on $b$, $A$ and $Z$ we show that the weak solution to the SDE is uniquely defined and Markov, we provide a representation of the transition probability density and we establish Holder regularity of the corresponding transition semigroup. The method used was essentially analytical and based on a version of Levi's method or parametrix method for a construction of the heat kernel for the associated Markov process, treated as a solution to the Kolmogorov backward differential equation. Finally, we will study some potential theory for the solution to the above equation in the case when the coordinates $Z^i$ of the noise $Z$ are i.i.d. scalar (symmetric) $\alpha$-stable processes, $ A(x)$ is bounded continuous and invertible, and $b\equiv 0$. In such case we provide an optimal two-sided estimate for the mean exit time of the process $X$ from an Euclidean ball. We also obtain sharp estimates for functions harmonic with respect to the process $X$ in balls, under the assumption that the Dirichlet exterior data are radial about the center.

• 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 - On some models of anomalous diffusion

  In this talk we will discuss several stochastic models of anomalous diffusion. In particular, we will focus on random processes subject to trapping effects and kinetic processes subject to random obstacles.
  For these different models, we will discuss the anomalous diffusive behaviour, the (non-)Markov property, the (non-local) PDE connections and simulation methods. Particular attention will be dedicated to their connection with the theory of scaling limits of Continuous Time Random Walks.

• Vanja Wagner - Nonhomogeneous boundary condition for spectral non-local operators

  The main interest of this talk will be the study of (semi-)linear nonlocal elliptic problems driven by spectral-type operators of the form $\psi(-L_{|D})$ in bounded $C^{1,1}$ domains $D\subset \mathbf R^d$. Here $\psi$ is a Bernstein function satisfying a weak scaling condition at infinity, and $L_{|D}$ is the generator of an appropriate killed unimodal L\'evy process. We will present the general framework that covers and extends the theory of the spectral and interpolated fractional Laplacian. The focus will be on the analysis of the non-homogeneous boundary condition via a weak $L^1$ trace-like boundary operator. This operator is formulated in terms of the Poisson potential with respect to the $d-1$ Hausdorff measure on $\partial D$. The methodology combines stochastic process techniques, potential theory, and spectral analysis, and expresses the boundary behaviour of the solution in terms of the renewal function and the distance to the boundary.

  The talk is based on a recent joint work with Ivan Biočić.