| Private Homepage | https://www.uni-muenster.de/Stochastik/en/Arbeitsgruppen/Mukherjee/ |
| Research Interests | Large deviations and stochastic analysis Directed polymers, stochastic PDEs, multiplicative chaos Stochastic homogenization, Hamilton-Jacobi equations Percolation, geometric group theory, C* algebras |
| Selected Publications | • Bazaes, R; Mukherjee, C; Ramirez, A; Saglietti, S Quenched and averaged large deviation rate functions for random walks in random environments: the impact of disorder. https://arxiv.org/abs/1906.05328 Vol. 2019, 2019 online • Mukherjee, C Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation. Annals of Applied Probability Vol. https://arxiv.org/abs/1706.09345, 2017 online • Mukherjee, C; Varadhan, SRS Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times. Communications on Pure and Applied Mathematics Vol. 73 (2), 2020, pp 350-383 online • Mukherjee, C; Varadhan, SRS Identification of the Polaron measure in strong coupling and the Pekar variational formula. Annals of Probability Vol. 48 (5), 2020, pp 2119-2144 online • Bröker, Y; Mukherjee, C Localization of the Gaussian multiplicative chaos in the Wiener space and the stochastic heat equation in strong disorder. Annals of Applied Probability Vol. 29 (6), 2019, pp 3745-3785 online • Berger, N; Mukherjee, C; Okamura, K. Quenched Large Deviations for Simple Random Walks on Percolation Clusters Including Long-Range Correlations. Communications in Mathematical Physics Vol. 358, 2018, pp 633–673 online • Bolthausen, E; König, W; Mukherjee, C Mean‐Field Interaction of Brownian Occupation Measures II: A Rigorous Construction of the Pekar Process. Communications on Pure and Applied Mathematics Vol. 70 (8), 2017, pp 1598-1629 online • Mukherjee, C Gibbs Measures on Mutually Interacting Brownian Paths under Singularities. Communications on Pure and Applied Mathematics Vol. 70 (12), 2017, pp 2366-2404 online • Mukherjee, C; Shamov, A; Zeitouni, O Weak and strong disorder for the stochastic heat equation and continuous directed polymers in d≥3. Electronic Communications in Probability Vol. 21, 2016, pp 1-12 online • Mukherjee, C; Varadhan, SRS Brownian occupation measures, compactness and large deviations. Annals of Probability Vol. 44 (6), 2016, pp 3934-3964 online |
| Topics in Mathematics Münster | T4: Groups and actions T7: Field theory and randomness T8: Random discrete structures and their limits T9: Multi-scale processes and effective behaviour |
| Current Publications | • Bazaes Rodrigo, Mukherjee Chiranjib, Ramírez Alejandro, Saglietti Santiago The effect of disorder on quenched and averaged large deviations for random walks in random environments: boundary behavior. https://arxiv.org/abs/2101.04606 Vol. na, 2021 online • Comets, F; Cosco, C; Mukherjee, C Renormalizing the Kardar-Parisi-Zhang equation in d≥3 in weak disorder. Journal of Statistical Physics Vol. 179, 2020, pp 713-728 online • Mukherjee, C; Varadhan, SRS Identification of the Polaron measure in strong coupling and the Pekar variational formula. Annals of Probability Vol. 48 (5), 2020, pp 2119-2144 online • Mukherjee, C; Varadhan, SRS Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times. Communications on Pure and Applied Mathematics Vol. 73 (2), 2020, pp 350-383 online • Bröker, Yannic and Mukherjee, Chiranjib Geometry of the Gaussian multiplicative chaos in the Wiener space. https://arxiv.org/abs/2008.04290 Vol. na, 2020 online • Bröker, Y; Comets, F; Cosco; C; Mukherjee, C; Shamov, A; Zeitouni; O KPZ equation in d ≥ 3 and construction of Gaussian multiplicative chaos in the Wiener space. , 2019 online • Adams, S; Mukherjee, C Commutative diagram of the Gross-Pitaevskii approximation. https://arxiv.org/abs/1911.09635 Vol. 2019, 2019 online • Altmeyer, G; Mukherjee, C On Null-homology and stationary sequences. https://arxiv.org/abs/1910.07378 Vol. 2019, 2019 online • Bazaes, R; Mukherjee, C; Ramirez, A; Saglietti, S Quenched and averaged large deviation rate functions for random walks in random environments: the impact of disorder. https://arxiv.org/abs/1906.05328 Vol. 2019, 2019 online |
| Current Projects | • EXC 2044 - T04: Groups and actions The study of symmetry and space through the medium of groups and their actions has long been a central theme in modern mathematics, indeed one that cuts across a wide spectrum of research within the Cluster. There are two main constellations of activity in the Cluster that coalesce around groups and dynamics as basic objects of study. Much of this research focuses on aspects of groups and dynamics grounded in measure and topology in their most abstract sense, treating infinite discrete groups as geometric or combinatorial objects and employing tools from functional analysis, probability, and combinatorics. Other research examines, in contrast to abstract or discrete groups, groups with additional structure that naturally arise in algebraic and differential geometry. online • EXC 2044 - T07: Field theory and randomness Quantum field theory (QFT) is the fundamental framework to describe matter at its smallest length scales. QFT has motivated groundbreaking developments in different mathematical fields: The theory of operator algebras goes back to the characterisation of observables in quantum mechanics; conformal field theory, based on the idea that physical observables are invariant under conformal transformations of space, has led to breakthrough developments in probability theory and representation theory; string theory aims to combine QFT with general relativity and has led to enormous progress in complex algebraic geometry, among others. online • EXC 2044 - T08: Random discrete structures and their limits Discrete structures are omnipresent in mathematics, computer science, statistical physics, optimisation and models of natural phenomena. For instance, complex random graphs serve as a model for social networks or the world wide web. Such structures can be descriptions of objects that are intrinsically discrete or they occur as an approximation of continuous objects. An intriguing feature of random discrete structures is that the models exhibit complex macroscopic behaviour, phase transitions in a wide sense, making the field a rich source of challenging mathematical questions. In this topic we will concentrate on three strands of random discrete structures that combine various research interests and expertise present in Münster. online • EXC 2044 - T09: Multiscale processes and effective behaviour Many processes in physics, engineering and life sciences involve multiple spatial and temporal scales, where the underlying geometry and dynamics on the smaller scales typically influence the emerging structures on the coarser ones. A unifying theme running through this research topic is to identify the relevant spatial and temporal scales governing the processes under examination. This is achieved, e.g., by establishing sharp scaling laws, by rigorously deriving effective scale-free theories and by developing novel approximation algorithms which balance various parameters arising in multiscale methods. online • GRK 3027: Rigorous Analysis of Complex Random Systems The Research Training Group is dedicated to educating mathematicians in the field of complex random systems. It provides a strong platform for the development of both industrial and academic careers for its graduate students. The central theme is a mathematically rigorous understanding of how probabilistic systems, modelled on a microscopic level, behave effectively at a macroscopic scale. A quintessential example for this RTG lies in statistical mechanics, where systems comprising an astronomical number of particles, upwards of 10^{23}, can be accurately described by a handful of observables including temperature and entropy. Other examples come from stochastic homogenisation in material sciences, from the behaviour of training algorithms in machine learning, and from geometric discrete structures build from point processes or random graphs. The challenge to understand these phenomena with mathematical rigour has been and continues to be a source of exciting research in probability theory. Within this RTG we strive for macroscopic representations of such complex random systems. It is the main research focus of this RTG to advance (tools for) both qualitative and quantitative analyses of random complex systems using macroscopic/effective variables and to unveil deeper insights into the nature of these intricate mathematical constructs. We will employ a blend of tools from discrete to continuous probability including point processes, large deviations, stochastic analysis and stochastic approximation arguments. Importantly, the techniques that we will use and the underlying mathematical ideas are universal across projects coming from completely different origin. This particular facet stands as a cornerstone within the RTG, holding significant importance for the participating students. For our students to be able to exploit the synergies between the different projects, they will pass through a structured and rich qualification programme with several specialised courses, regular colloquia and seminars, working groups, and yearly retreats. Moreover, the PhD students will benefit from the lively mathematical community in Münster with a mentoring programme and several interaction and networking activities with other mathematicians and the local industry. • EXC 2044 - C2: Multi-scale phenomena and macroscopic structures In multi-scale problems, geometry and dynamics on the micro-scale influence structures on coarser scales. In this research unit we will investigate and analyse such structural interdependence based on topological, geometrical or dynamical properties of the underlying processes. We are interested in transport-dominated processes, such as in the problem of how efficient a fluid can be stirred to enhance mixing, and in the related analytical questions. A major concern will be the role of molecular diffusion and its interplay with the stirring process. High Péclet number flow in porous media with reaction at the surface of the porous material will be studied. Here, the flowinduces pore-scale fluctuations that lead to macroscopic enhanced diffusion and reaction kinetics. We also aim at understanding advection-dominated homogenisation problems in random regimes. We aim at classifying micro-scale geometry or topology with respect to the macroscopic behaviour of processes considered therein. Examples are meta material modelling and the analysis of processes in biological material. Motivated by network formation and fracture mechanics in random media, we will analyse the effective behaviour of curve and free-discontinuity energies with stochastic inhomogeneity. Furthermore, we are interested in adaptive algorithms that can balance the various design parameters arising in multi-scale methods. The analysis of such algorithms will be the key towards an optimal distribution of computational resources for multi-scale problems. Finally, we will study multi-scale energy landscapes and analyse asymptotic behaviour of hierarchical patterns occurring in variational models for transportation networks and related optimal transport problems. In particular, we will treat questions of self-similarity, cost distribution, and locality of the fine-scale pattern. We will establish new multilevel stochastic approximation algorithms with the aim of numerical optimisation in high dimensions. online • EXC 2044 - C3: Interacting particle systems and phase transitions The question of whether a system undergoes phase transitions and what the critical parameters are is intrinsically related to the structure and geometry of the underlying space. We will study such phase transitions for variational models, for processes in random environments, for interacting particle systems, and for complex networks. Of special interest are the combined effects of fine-scalerandomly distributed heterogeneities and small gradient perturbations. We aim to connect different existing variational formulations for transportation networks, image segmentation, and fracture mechanics and explore the resulting implications on modelling, analysis, and numerical simulation of such processes. We will study various aspects of complex networks, i.e. sequences of random graphs (Gn)n∈N, asking for limit theorems as n tends to infinity. A main task will be to broaden the class of networks that can be investigated, in particular, models which include geometry and evolve in time. We will study Ising models on random networks or with random interactions, i.e. spin glasses. Fluctuations of order parameters and free energies will be analysed, especially at the critical values where the system undergoes a phase transition. We will also investigate whether a new class of interacting quantum fields connected with random matrices and non-commutative geometry satisfies the Osterwalder-Schrader axioms. Further, we will study condensation phenomena, where complex network models combine the preferential attachment paradigm with the concept of fitness. In the condensation regime, a certain fraction of the total mass dynamically accumulates at one point, the condensate. The aim is a qualitative and quantitative analysis of the condensation. We willalso explore connections to structured population models. Further, we will study interacting particle systems on graphs that describe social interaction or information exchange. Examples are the averaging process or the Deffuant model. We will also analyse asymmetric exclusion processes (ASEP) on arbitrary network structures. An interesting aspect will be how these processes are influenced by different distribution mechanisms of the particles at networks nodes. If the graph is given by a lattice, we aim to derive hydrodynamic limits for the ASEP with jumps of different ranges for multiple species, and for stochastic interactingmany-particle models of reinforced random walks. Formally, local cross-diffusion syste ms are obtained as limits of the classical multi-species ASEP and of the many-particle random walk. We will compare the newly resulting limiting equations and are interested in fluctuations, pattern formation, and the long-time behaviour of these models on the microscopic and the macroscopic scale. Further, we will analyse properties of the continuous directed polymer in a random environment. online | chiranjib dot mukherjee at uni-muenster dot de |
| Phone | +49 251 83-33772 |
| FAX | +49 251 83-32712 |
| Room | 130.012 |
| Secretary | Sekretariat Stochastik Frau Claudia Giesbert Telefon +49 251 83-33792 Fax +49 251 83-32712 Zimmer 120.002 |
| Address | Prof. Dr. Chiranjib Mukherjee Institut für Mathematische Stochastik Fachbereich Mathematik und Informatik der Universität Münster Orléans-Ring 10 48149 Münster |
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