Stochastic Partial Differential Equations and Related Fields

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.

Preface.- Longer Contributions: V. Bogachev, Stationary Fokker-Planck-Kolmogorov equations.- M. Fukushima, Liouville property of harmonic functions of finite energy for Dirichlet forms.- B. Gess, Regularization and well-posedness by noise for ordinary and partial differential equations.- M. Gubinelli, N. Perkowski, An introduction to singular SPDEs.-  N. Perkowski, M. Gubinelli, Probabilistic approach to the stochastic Burgers equation.- G. Da Prato, Fokker-Planck equations in Hilbert spaces.- Stochastic Partial Differential Equations and Regularity structures: Z. Brzezniak, G. Dhariwal, J. Hussain, M. Mariani, Stochastic and deterministic constrained partial differential equations.- P. Coupek, B. Maslowski, J. Šnupárková, SPDEs with Volterra Noise.- R. Dalang, Hitting probabilities for systems of stochastic PDEs: an overview.- C. Denis, T. Funaki, S. Yokoyama, Curvature motion perturbed by a direction-dependent colored noise.- N.  Krylov, E. Priola, Poisson stochastic process and basic Schauder and Sobolev estimates in the theory of parabolic equations.- H. D. Luu, M. J. Garrido-Atienza, B. Schmalfuss, Dynamics of SPDE driven by a small fractional Brownian motion with Hurst parameter larger than ½.- C. Marinelli, L. Scarpa, On the well-posedness of SPDEs with singular drift in divergence form.- L. J. de Naurois, A. Jentzen, T. Welti, Lower bounds for weak approximation errors for spatial spectral Galerkin approximations of stochastic wave equations.- J. M. Tölle, Estimates for nonlinear stochastic partial differential equations with gradient noise via Dirichlet forms.- W. Yan, J. Duan, Random data Cauchy problem for some dispersive equations.- Lorenzo Zambotti, SPDEs and Renormalisation.- D. Zhang, Recent progress on stochastic nonlinear Schrödinger equations.- Stochastic Analysis including geometric aspects: V. Barbu, Generalized Solutions to Nonlinear Fokker-Planck Equations with Linear Drift.- Y. Bruned, I. Chevyrev, P. Friz, Examples of Renormalized SDEs.- K.D. Elworthy, Generalised Weitzenböck formulae for differential operators in Hörmander form.- M. Hofmanova, On the Rough Gronwall Lemma and its Applications.- X.-M. Li, Doubly Damped Stochastic Parallel translations and Hessian formulas.- M. Scheutzow, I. Vorkastner, Synchronization, Lyapunov exponents and stable manifolds for random dynamical Systems.- S. Shaposhnikov, Nonlinear Fokker-Planck-Kolmogorov equations for measures.- F.-Y. Wang, Coupling by Change of Measure, Harnack Inequality and Hypercontractivity.- X. Zhang, Multidimensional Singular Stochastic Differential Equations.- Dirichlet forms, Markov Processes and Potential theory: L. Beznea, I. Cimpean, L. Beznea, I. Cimpean. Invariant, super and quasi-martingale functions of a Markov process.- Z.-Q. Chen, T.  Kumagai, J. Wang, Mean value inequalities for jump processes.- X. Chen, Z.-M. Ma, X. Peng, Positivity preserving semigroups and positivity preserving coercive forms.- N. Jacob, J. Harris, Some Thoughts andInvestigations on Densities of One-Parameter Operator Semi-groups.- H. Kawabi, Strong uniqueness of Dirichlet operators related to stochastic quantization under exponential interactions in one-dimensional infinite volume.- V. Knopova, R. Schilling, A Probabilistic proof of the breakdown of Besov regularity in L-shaped domains.- T. Masayoshi, Symmetric Markov Processes with Tightness Property.- Applications including Mathematical Physics: J. Chen, M. Hinz, A. Teplyaev, From non-symmetric particle systems to non-linear PDEs on fractals.- Y. Kozitsky, Equilibrium States, Phase Transitions and Dynamics in Quantum Anharmonic Crystals.- X. Liu, B. Zegarlinski, On Continuous Coding.- H. Osada, Infinite-dimensional Stochastic Differential Equations with Symmetry.- R. Zhu, X. Zhu, Recent progress on the Dirichlet forms associated with stochastic quantization Problems.