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This conference proceeding contains 27 peer-reviewed invited papers from leading experts as well as young researchers all over the world in the related fields that Professor Fukushima has made important contributions to. These 27 papers cover a wide range of topics in probability theory, ranging from Dirichlet form theory, Markov processes, heat kernel estimates, entropy on Wiener spaces, analysis on fractal spaces, random spanning tree and Poissonian loop ensemble, random Riemannian geometry, SLE, space-time partial differential equations of higher order, infinite particle systems, Dyson model, functional inequalities, branching process, to machine learning and Hermitizable problems for complex matrices. Researchers and graduate students interested in these areas will find this book appealing.

The book covers the latest research in the areas of mathematics that deal the properties of partial differential equations and stochastic processes on spaces in connection with the geometry of the underlying space. Written by experts in the field, this book is a valuable tool for the advanced mathematician.

Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.

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.

We introduce conditions on the symmetric and skew-symmetric parts of timedependent, local, regular forms that imply a parabolic Harnack inequality for appropriate weak solutions of the associated heat equation, under natural assumptions on the underlying space. In particular, these local weak solutions are locally bounded and Holder continuous. Precise two-sided heat kernel estimates are deo rived from this parabolic Harnack inequality. For Dirichlet forms satisfying our conditions we prove a scale-invariant boundary Harnack principle in inner uniform domains. Inner uniformity is a condition on the boundary of the domain that is described solely in terms of the intrinsic length metric of the domain. In addition, we show that the Martin boundary of an inner uniform domain is homeomorphic to the boundary of the domain with respect to its completion in the inner distance. The main result of this work are two-sided Gaussian bounds for Dirichlet heat kernels corresponding to (non- )symmetric, local, regular Dirichlet forms. These bounds hold in domains that satisfy the inner uniformity condition. The proof uses the parabolic Harnack inequality and the boundary Harnack principle described above, as well as the Doob h-transform technique. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa, A. Ancona, P. Gyrya, L. Saloff-Coste and K.-T. Sturm.