Stochastic Partial Differential Equations and Related Fields

Description

This Festschrift contains five research surveys and thirty-four shorter contributions by participants of the conference ''Stochastic Partial Differential Equations and Related Fields'' hosted by the Faculty of Mathematics at Bielefeld University, October 10–14, 2016. The conference, attended by more than 140 participants, including PostDocs and PhD students, was held both to honor Michael Röckner's contributions to the field on the occasion of his 60th birthday and to bring together leading scientists and young researchers to present the current state of the art and promising future developments. Each article introduces a well-described field related to Stochastic Partial Differential Equations and Stochastic Analysis in general. In particular, the longer surveys focus on Dirichlet forms and Potential theory, the analysis of Kolmogorov operators, Fokker–Planck equations in Hilbert spaces, the theory of variational solutions to stochastic partial differential equations, singular stochastic partial differential equations and their applications in mathematical physics, as well as on the theory of regularity structures and paracontrolled distributions. The numerous research surveys make the volume especially useful for graduate students and researchers who wish to start work in the above-mentioned areas, or who want to be informed about the current state of the art. Front Matter ....Pages i-xx Front Matter ....Pages 1-1 Stationary Fokker–Planck–Kolmogorov Equations (Vladimir I. Bogachev)....Pages 3-24 Liouville Property of Harmonic Functions of Finite Energy for Dirichlet Forms (Masatoshi Fukushima)....Pages 25-42 Regularization and Well-Posedness by Noise for Ordinary and Partial Differential Equations (Benjamin Gess)....Pages 43-67 An Introduction to Singular SPDEs (Massimiliano Gubinelli, Nicolas Perkowski)....Pages 69-99 Fokker–Planck Equations in Hilbert Spaces (Giuseppe Da Prato)....Pages 101-129 Front Matter ....Pages 131-131 Stochastic and Deterministic Constrained Partial Differential Equations (Zdzisław Brzeźniak, Gaurav Dhariwal, Javed Hussain, Mauro Mariani)....Pages 133-146 SPDEs with Volterra Noise (Petr Čoupek, Bohdan Maslowski, Jana Šnupárková)....Pages 147-158 Hitting Probabilities for Systems of Stochastic PDEs: An Overview (Robert C. Dalang)....Pages 159-176 Curvature Motion Perturbed by a Direction-Dependent Colored Noise (Clément Denis, Tadahisa Funaki, Satoshi Yokoyama)....Pages 177-200 Poisson Stochastic Process and Basic Schauder and Sobolev Estimates in the Theory of Parabolic Equations (Short Version) (N. V. Krylov, E. Priola)....Pages 201-211 Dynamics of SPDEs Driven by a Small Fractional Brownian Motion with Hurst Parameter Larger than 1/2 (L. H. Duc, M. J. Garrido-Atienza, B. Schmalfuß)....Pages 213-224 On the Well-Posedness of SPDEs with Singular Drift in Divergence Form (Carlo Marinelli, Luca Scarpa)....Pages 225-235 Lower Bounds for Weak Approximation Errors for Spatial Spectral Galerkin Approximations of Stochastic Wave Equations (Ladislas Jacobe de Naurois, Arnulf Jentzen, Timo Welti)....Pages 237-248 Estimates for Nonlinear Stochastic Partial Differential Equations with Gradient Noise via Dirichlet Forms (Jonas M. Tölle)....Pages 249-262 Random Data Cauchy Problem for Some Dispersive Equations (Wei Yan, Jinqiao Duan)....Pages 263-269 SPDEs and Renormalisation (Lorenzo Zambotti)....Pages 271-277 Recent Progress on Stochastic Nonlinear Schrödinger Equations (Deng Zhang)....Pages 279-289 Front Matter ....Pages 291-291 Generalized Solutions to Nonlinear Fokker–Planck Equations with Linear Drift (Viorel Barbu)....Pages 293-302 Examples of Renormalized SDEs (Y. Bruned, I. Chevyrev, P. K. Friz)....Pages 303-317 Generalised Weitzenböck Formulae for Differential Operators in Hörmander Form (K. D. Elworthy)....Pages 319-331 On the Rough Gronwall Lemma and Its Applications (Martina Hofmanová)....Pages 333-344 Doubly Damped Stochastic Parallel Translations and Hessian Formulas (Xue-Mei Li)....Pages 345-357 Synchronization, Lyapunov Exponents and Stable Manifolds for Random Dynamical Systems (Michael Scheutzow, Isabell Vorkastner)....Pages 359-366 Nonlinear Fokker–Planck–Kolmogorov Equations for Measures (Stanislav V. Shaposhnikov)....Pages 367-379 Coupling by Change of Measure, Harnack Inequality and Hypercontractivity (Feng-Yu Wang)....Pages 381-389 Multidimensional Singular Stochastic Differential Equations (Xicheng Zhang)....Pages 391-403 Front Matter ....Pages 405-405 Invariant, Super and Quasi-martingale Functions of a Markov Process (Lucian Beznea, Iulian Cîmpean)....Pages 407-420 Mean Value Inequalities for Jump Processes (Zhen-Qing Chen, Takashi Kumagai, Jian Wang)....Pages 421-437 Positivity Preserving Semigroups and Positivity Preserving Coercive Forms (Xian Chen, Zhi-Ming Ma, Xue Peng)....Pages 439-449 Some Thoughts and Investigations on Densities of One-Parameter Operator Semi-groups (James Harris, Niels Jacob)....Pages 451-460 Strong Uniqueness of Dirichlet Operators Related to Stochastic Quantization Under Exponential Interactions in One-Dimensional Infinite Volume (Hiroshi Kawabi)....Pages 461-471 A Probabilistic Proof of the Breakdown of Besov Regularity in L-Shaped Domains (Victoria Knopova, René L. Schilling)....Pages 473-488 Symmetric Markov Processes with Tightness Property (Masayoshi Takeda)....Pages 489-499 Front Matter ....Pages 501-501 From Non-symmetric Particle Systems to Non-linear PDEs on Fractals (Joe P. Chen, Michael Hinz, Alexander Teplyaev)....Pages 503-513 Probabilistic Approach to the Stochastic Burgers Equation (Massimiliano Gubinelli, Nicolas Perkowski)....Pages 515-527 Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals (Yuri Kozitsky)....Pages 529-538 On Continuous Coding (Xing Liu, Bogusław Zegarliński)....Pages 539-547 Infinite-Dimensional Stochastic Differential Equations with Symmetry (Hirofumi Osada)....Pages 549-559 Recent Progress on the Dirichlet Forms Associated with Stochastic Quantization Problems (Rongchan Zhu, Xiangchan Zhu)....Pages 561-571 Back Matter ....Pages 573-574