# Prof. Dr. Chiranjib Mukherjee, Institut für Mathematische Stochastik

Member of Mathematics MünsterInvestigator in Mathematics Münster

Investigator in Mathematics Münster

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 | • Mukherjee, C; Varadhan, SRS Brownian occupation measures, compactness and large deviations. Annals of Probability Vol. 44 (6), 2016, pp 3934-3964 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• Comets, F; Cosco, C; Mukherjee, C Space-time fluctuation of the Kardar-Parisi-Zhang equation in d≥3 and the Gaussian free field. https://arxiv.org/abs/1905.03200 Vol. 2019, 2019 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; 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• 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• 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 |

Project membershipMathematics Münster | C: Models and ApproximationsC2: Multi-scale phenomena and macroscopic structures C3: Interacting particle systems and phase transitions |

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 - 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 |

E-Mail | chiranjib dot mukherjee at uni-muenster dot de |

Phone | +49 251 83-33772 |

FAX | +49 251 83-32712 |

Room | 130.012 |

Secretary | Sekretariat Kollwitz Frau Anita Kollwitz Telefon +49 251 83-33770 Fax +49 251 83-32712 Zimmer 130.030 |

Address | Prof. Dr. Chiranjib Mukherjee Institut für Mathematische Stochastik Fachbereich Mathematik und Informatik der Universität Münster Orléans-Ring 10 48149 Münster |

Diese Seite editieren |