Workshop START 2023: STochastic Analysis and Related Topics
Workshop START 2023: STochastic Analysis and Related Topics
Time:
Thursday 23^{rd} and Friday 24^{th} of November 2023
Venue:
TU Dresden, Mathematics
Building Z21: Zellescher Weg 25, 01217 Dresden (Map)
Room 250 (Map)
Organizers:
Anita Behme, Martin KellerRessel, René Schilling
______________________________________________________________________________________
Download Picture (2,2 MB)
Participants & Talks:
Schedule

Sebastian Andres (Braunschweig)
Heat kernel fluctuations and quantitative homogenizations for the onedimensional Bouchaud trap model
Abstract: It is wellknown that stochastic processes on fractal spaces or in certain random media exhibit anomalous heat kernel behaviour. One manifestation of such irregular behaviour is the presence of fluctuations in the short or longtime asymptotics of the ondiagonal heat kernel. In this talk we discuss such heat kernel fluctuations for the onedimensional Bouchaud trap model, that is a random walk in a random medium given by a landscape of traps which retain the walk for some amount of time. It has its origins in the statistical physics literature, where it was proposed as a simple effective model for the dynamics of spinglasses on certain timescales. We also present some quantitative homogenization results for the model, these include both quenched and annealed BerryEsseentype theorems, as well as a quantitative quenched local limit theorem.
This talk is based on a joint work with David Croydon (Kyoto) and Takashi Kumagai (Waseda). 
Giacomo Ascione (Napoli)
Coupling Plateaux and Jumps: the Undershooting of Subordinators and the Corresponding SemiMarkov Processes
Abstract: In recent years, there has been growing attention on general semiMarkov processes and their governing equations. Among them, one can consider semiMarkov processes obtained by applying a timechange to Feller processes by means of an inverse subordinator independent of it. Such processes have been widely studied and their link with timenonlocal equations (for instance, timefractional ones) has been exploited. On the other hand, the composition of a Feller process with an independent subordinator leads to a new Feller process whose generator is obtained by means of Bochner subordination, and thus is a nonlocal operator. Here we construct another class of timechanged processes. Precisely, we consider the composition of a Feller process with the undershooting process of an independent subordinator, i.e. the left limits of the composition of the subordinator with its inverse. We first show that such processes are semiMarkov and then we proceed with the study of their Kolmogorov equations. However, since there is a form of coupling between plateaux and jumps of the undershooting process, such equations involve an integrodifferential operator which couples the nonlocality in both space and time. We determine a class of functions on which such an operator can be applied and then we use this class to prove that the process provides a stochastic representation of solutions of an integrodifferential equation. Furthermore, we exploit an application of this theory to subdiffusive Black and Scholes models.
This is an ongoing joint work with Enrico Scalas from the Sapienza University of Rome, Bruno Toaldo from the University of Turin, and Lorenzo Torricelli from the University of Bologna. 
Justin Baars (Amsterdam)
Delayed Hawkes birthdeath processes
Abstract: We introduce a variant of the Hawkesfed birthdeath process, in which the conditional intensity does not increase at arrivals, but at departures from the system. Since arrivals cause excitation after a delay equal to their lifetimes, we call this a delayed Hawkes process. We introduce a general family of models admitting a cluster representation containing the Hawkes, delayed Hawkes and ephemerally selfexciting processes as special cases. For this family of models, as well as their nonlinear extensions, we prove existence, uniqueness and stability. Our family of models satisfies the same FCLT as the classical Hawkes process; however, we describe a scaling limit for the delayed Hawkes process in which sojourn times are stretched out by a factor $\sqrt T$, after which time gets contracted by a factor $T$. This scaling limit highlights the effect of sojourntime dependence. The cluster representation renders our family of models tractable, allowing for transform characterisation by a fixedpoint equation and for an analysis of heavytailed asymptotics. In the Markovian case, for a network of delayed Hawkes birthdeath processes, an explicit recursive procedure is presented to calculate the $d$thorder moments analytically. Finally, we compare the delayed Hawkes process to the regular Hawkes process in the stochastic ordering, which enables us to describe stationary distributions and heavytraffic behaviour. 
Julio BackhoffVeraguas (Uni Wien)
Bass Martingales: existence, duality, and their properties
Abstract: Motivated by robust mathematical finance, and also taking inspiration from the field of Optimal Transport, we ask: What is the martingale, with prescribed initial and terminal marginal distributions, which is closest to Brownian motion?
Under suitable assumptions the answer to this question, in any dimension, is provided by 'Bass martingales', ie. orderpreserving and martingalepreserving stretchings of an underlying Brownian motion. In this talk we discuss the properties of Bass martingales, their existence, and the duality theory required to study them. Based on joint work with Beiglböck, Schachermayer and Tschiderer. 
Adam Jakubowski (Torun)
Probability on submetric spaces
In 1953 Yu. Prohorov published a paper on weak convergence of probability measures on metric spaces, bringing a new, extended context to the Invariance Principle proved by M. Donsker two years earlier.
Prohorov’s formalism, publicised in books by K.R. Parthasarathy and P. Billinsley, established the equivalence of the notion of convergence in law of stochastic processes and the weak convergence of their distributions. This point of view is completely justified in metric spaces,especially in Polish spaces. It is, however, much less satisfactory in nonmetric spaces, as was shown by an example due to X. Fernique, given long time ago. We show that in a large class of submetric spaces there exists a stronger mode of convergence, coinciding with the weak convergence on metric spaces, and much more suitable for needs of contemporary theory of stochastic partial differential equations. A submetric space is a topological space (X, τ ) admitting a continuous metrics d that in turn determines a metric topology τd ⊂ τ (where this inclusion is in general strict). As a standard (and the simplest) example may serve a separable Hilbert space equipped with the weak topology. 
Zhezhe Jiao (TU Dresden)
Emergence of heavy tails in stochastic gradient descent
Abstract: tba. 
Alex Kulik (Wrocław)
Intrinsic geometry and analysis of multivariate Lévy processes
Abstract: Based on a multivariate analogue of the Pruitte function for a Lévy process, we introduce a family of concentration norms which appear to be quite a precise tool for describing concentration properties of the law of the Léevy process on a given time interval and estimating its distribution density together with derivatives. The estimates based on concentration norms appear to be naturally applicable within further constructions of the parametrix method for Lévydriven SDEs with spatially heterogeneous Lévy noises. 
Khoa Lê (Leeds)
JohnNirenberg inequality for BMO processes
Abstract: The JohnNirenberg inequality provides exponential integrability for stochastic processes with bounded mean oscillation (BMO). Although the result has been known since 70's thanks to several works of Garsia and Stroock, its applications seem to be limited. We will report some new applications of such result in studying stochastic differential equations. 
Víctor Manuel Rivero Mercado (CIMAT)
Williams' path decomposition for ddimensional selfsimilar Markov processes
Abstract: The classical result of Williams states that a Brownian motion with positive drift A and issued from the origin is equal in law to a Brownian motion with unit negative drift, A, run until it hits a negative threshold, whose depth below the origin is independently and exponentially distributed with parameter 2A, after which it behaves like a Brownian motion conditioned never to go below the aforesaid threshold (i.e. a Bessel3 process, or equivalently a Brownian motion conditioned to stay positive, relative to the threshold).
In this talk we will consider the analogue of Williams' path decomposition for a general ddimensional selfsimilar Markov process (ssMp). In essence, Williams' path decomposition in the setting of a ssMp follows directly from an analogous decomposition for Markov additive processes (MAPs). The latter class are intimately related to the former via a spacetime transform known as the LampertiKiu transform. As a key feature of our proof of Williams' path decomposition, will obtain the analogue of Silverstein's duality identity for the excursion occupation measure for general Markov additive processes (MAPs). 
Aleks Mijatovic (Warwick)
Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability
Abstract: In this talk we quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field. Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applicable to general continuous semimartingales. Time permitting, I will discuss the main ideas behind the proofs in the talk. This is joint work with Miha Bresar (Warwick) and Andrew Wade (Durham). 
Zbigniew Palmowski (Wroclaw)
Stationary states for Lévy processes with partial resetting
Abstract 
Alexander Schnurr (Siegen)
From Markov Processes to Semimartingales
Abstract: In the development of stochastic integration and the theory of semimartingales, Markov processes have been a constant source of inspiration. Despite this historical interweaving, it turned out that semimartingales should be considered the `natural' class of processes for many concepts first developed in the Markovian framework.
As an example, stochastic differential equations have been invented as a tool to study Markov processes but nowadays are treated separately in the literature. Moreover, the killing of processes has been known for decades before it made its way to the theory of semimartingales most recently.
We describe, when these and other important concepts have been invented in the theory of Markov processes and how they were transferred to semimartingales. 
Thomas Simon (Lille)
On the αSun distribution
Abstract: The αSun distribution arises as a limit law of a certain stochastic scheme introduced by Greenwood and Hoogmiestra, interpolating between sum and maximum. We will present an analytical study of this distribution, including visual shape, exact asymptotic behaviour and infinite divisibility properties. The connection with exponential functionals of subordinators will be emphasized, and we will raise several open questions in this respect. 
Bruno Toaldo (Turin)
From SemiMarkov Evolutions to Scattering Transport and Superdiffusions
Abstract: We consider random evolutions driven by a class of semiMarkov processes. The expectation of such evolutions is shown to solve abstract Cauchy problems, which is nonlocal in time and space. Further, the abstract telegraph (damped waves) equation is generalized to this semiMarkov setting. Particular attention is devoted to semiMarkov models of scattering transport processes which
can be represented through these evolutions. It turns out that the Cauchy problem in this case is a direct (nonlocal) generalization of the linear Boltzmann equation. In particular, random motions with infinite mean scattering times are considered and their scaling limit is proved to converge to a superdiffusive process.