Program

Let \(Z\) be an isotropic stable process in the Euclidean space. The process \(Z\) is killed upon exiting an open set \(D\) and the killed process is then subordinated by an independent \(\gamma\)-stable subordinator, \(0<\gamma <1\). The resulting process is a Hunt process in \(D\). In this talk, I will discuss several potential theoretical properties of this process such as Harnack inequality for nonnegative harmonic functions, the Carleson estimate, Green function and jumping kernel estimates in smooth sets \(D\), and in particular, the boundary Harnack principle. Surprisingly, it turns out the BHP holds only if \(1/2<\gamma<1\). This is joint work with Panki Kim and Renming Song.

Fractional and anomalous diffusions have a long history. The terms fractional and anomalous have been considered with different meaning and in different contexts. By fractional diffusion we mean a diffusion in a medium with fractional dimension (fractals, for instance) whereas, by anomalous diffusions, according to the most significant literature, we refer to a motion whose mean squared displacement is proportional to a power of time. In this context we usually have the characterization given in terms of subdiffusion/superdiffusion or normal diffusion. Our aim is to pay exclusive attention to the probabilistic models for anomalous dynamics (not necessarily anomalous diffusions). The involved processes are guided by fractional (in time/space) equations and are obtained through random time changes.

We introduce a precise definition of delayed and rushed processes and provide some examples which are, in some cases, counter-intuitive.

We consider time changes given by subordinators and their inverse processes. Our analysis shows that, quite surprisingly, inverse processes are not necessarily leading to delayed processes.

We prove that a space inversion property of a standard Markov process X implies the existence of a Kelvin transform of X-harmonic, excessive and operator–harmonic functions and that the inversion property is inherited by Doob h-transforms. We determine new classes of processes having space inversion properties amongst those satisfying the time inversion property. For these processes, some explicit inversions, which are often not the spherical ones, and excessive functions are given explicitly. Some examples are treated in details.

I will consider a class of non-local equations, assuming that their solutions have a global maximum, and discuss estimates on their position. Such points are of special interest as they describe where hot-spots settle on the long run or, in a probabilistic description, the region around which paths typically concentrate. I will address, on the one hand, the interplay of competing parameters whose compromise the resulting location is, and the impact of geometric constraints, on the other.

References
[1] A. Biswas and J. Lőrinczi. Universal constraints on the location of extrema of eigenfunctions of non-local Schrödinger operators, arXiv:1710.11596, 2017 (under review)
[2] A. Biswas and J. Lőrinczi. Maximum principles and Aleksandrov-Bakelman-Pucci type estimates for non-local Schrödinger equations with exterior conditions, arXiv:1711.09267, 2017 (under review)
[3] A. Biswas and J. Lőrinczi. Maximum principles for time-fractional Cauchy problems with spatially non-local components, arXiv:1801.02349, 2018 (under review)
[4] A. Biswas and J. Lőrinczi. Ambrosetti-Prodi type results for Dirichlet problems of the fractional Laplacian, arXiv:1803.08540, 2018 (under review)

The Generalized Elastic Model accounts for the dynamics of several physical systems, such as polymers, fluctuating interfaces, growing surfaces, membranes, proteins and file systems among others. We derive the fractional stochastic equation governing the motion of a probe particle (tracer) in such kind of systems. This Langevin equation involves the use of fractional derivative in time and satisfies the Fluctuation-Dissipation relation, it goes under the name of Fractional Langevin Equation. Within this framework the spatial correlations appearing in the Generalized Elastic Model are translated into time correlations described by the fractional derivative together with the space-time correlations of the fractional Gaussian noise. We derive the exact scaling analytical form of several physical observables such as structure factors, roughness and mean square displacement. Special attention will be devoted to the dependence on initial conditions and linear-response relations in the case of an applied potential.

Cellular signaling is regulated by biochemical interactions that are ultimately controlled by molecular diffusion. Recent advances in fluorescence microscopy have allowed the visualization of single molecules in living cells at unprecedented spatiotemporal resolution, revealing that the heterogeneity of the cellular environment produces exotic molecular motions that deviate from Brownian behavior [1]. These findings have stimulated new questions about the mechanisms generating these phenomena, as well as regarding their implications for cell biology.

In this context, we have studied a transmembrane receptor involved in the capture of pathogens, which motion exhibits anomalous diffusion with signatures of weak ergodicity breaking [2]. Through the study of receptor mutants, we have been able to correlate the receptors motion to its molecular structures, lateral organization and interactions, thus establishing a link between nonergodicity and biological function. In addition, we have quantitatively interpreted the receptor dynamics through a stochastic model of random motion with random diffusivity on scale-free media [3,4], and we are attempting to gain further insight into the molecular causes of this complex diffusion.

Our work highlights the role of heterogeneity in cell membranes and proposes a connection with function regulation. In addition, our models offer a theoretical framework to interpret anomalous transport in complex media, such as those found, e.g., in soft condensed matter, geology, and ecology.


References

[1]C. Manzo, and M. F. Garcia-Parajo, Rep. Prog. Phys. 78:124601 (2015).
[2]C. Manzo, et al., Phys. Rev. X 5:011021 (2015).
[3]P. Massignan, et al., Phys. Rev. Lett. 112:150603 (2014).
[4]C. Charalambous, et al., Phys. Rev. E 95:032403 (2017).

Recent developments in modeling of different physical and biological systems undoubtedly confirmed that fractional calculus is not just an exotic mathematical theory, as it might appeared at the dawn of its emergence. The present work aims to demonstrate this through physical examples of diffusion and quantum motion in two-dimensional comb-like structures, leading to fractional calculus description of the corresponding processes. Comb geometry is one of the most simple paradigm of a two-dimensional structure, where anomalous diffusion can be realized. The comb model, for the first time, has been applied to investigate anomalous diffusion in low-dimensional percolation clusters, and nowadays it has many applications in description of transport properties in porous dielectrics and low dimensional composites, turbulent diffusion on a comb, tracer dynamics in subsurface aquifers, and anomalous diffusion in spiny dendrites, to name but a few. In this presentation we will show how the time fractional diffusion and Schrödinger equations can be derived by projection of the two dimensional comb dynamics in the one-dimensional configuration space. The anomalous character of the corresponding process can be described by the fact that the particle which moves along the main backbone channel can get trapped in the fingers of the comb. We will also show how the anomalous diffusion along the main backbone depends on the fractal structure of the comb, resulting in natural appearance of fractional derivatives in the corresponding equations for the probability density function. The connection between the anomalous diffusion exponent and the fractal dimension of the comb/mesh structures has been recently experimentally observed in the anomalous transport through porous structurally inhomogeneous media.

We consider a one-dimensional nonlinear diffusion equation with nonlocal time, i.e. the temporal derivative is calculated via the Riemann-Liouville operator. This problem set on a half-line models moisture percolation in certain porous media such as construction materials.

The efficient and convergent numerical method can be constructed as follows. We cast our equation into the self-similar form with the use of the Erdelyi-Kober operator. Then, by a suitable transformation the free-boundary problem is transformed into the initial-value one. Lastly, the resulting integro-differential equation can be written as a nonlinear Volterra equation. In this way, a nonlinear PDE with free boundary can be solved by finding a solution of an ordinary Volterra equation. This quickens the calculations tremendously.

We also prove that a certain family of numerical schemes is convergent to the solution of the aforementioned Volterra equation. The main difficulty lies in the non-Lipschitzian character of the governing nonlinearity. In that case we cannot use the classical theory and have to proceed in other ways. In the talk we will describe the details of the proof.