Michael Röckner

Introduction to the theory of (non-symmetric) Dirichlet forms

0 Introduction.- I Functional Analytic Background.- 1 Resolvents, semigroups, generators.- 2 Coercive bilinear forms.- 3 Closability.- 4 Contraction properties.- 5 Notes/References.- II Examples.- 1 Starting point: operator.- 2 Starting point: bilinear form - finite dimensional case.- 3 Starting point: bilinear form - infinite dimensional case.- 4 Starting point: semigroup of kernels.- 5 Starting point: resolvent of kernels.- 6 Notes/References.- III Analytic Potential Theory of Dirichlet Forms.- 1 Excessive functions and balayage.- 2 ?-exceptional sets and capacities.- 3 Quasi-continuity.- 4 Notes/References.- IV Markov Processes and Dirichlet Forms.- 1 Basics on Markov processes.- 2 Association of right processes and Dirichlet forms.- 3 Quasi-regularity and the construction of the process.- 4 Examples of quasi-regular Dirichlet forms.- 5 Necessity of quasi-regularity and some probabilistic potential theory.- 6 One-to-one correspondences.- 7 Notes/References.- V Characterization of Particular Processes.- 1 Local property and diffusions.- 2 A new capacity and Hunt processes.- 3 Notes/References.- VI Regularization.- 1 Local compactification.- 2 Consequences - the transfer method.- 3 Notes/References.- A Some Complements.- 1 Adjoint operators.- 2 The Banach/Alaoglu and Banach/Saks theorems.- 3 Supplement on Ray resolvents and right processes.

Stochastic differential equations in infinite dimensions: solutions via Dirichlet forms

SummaryUsing the theory of Dirichlet forms on topological vector spaces we construct solutions to stochastic differential equations in infinite dimensions of the type

Classical Dirichlet forms on topological vector spaces—Closability and a Cameron-Martin formula

dX_t = dW_t + \beta (X_t )dt

Harnack and functional inequalities for generalized Mehler semigroups

for possibly very singular drifts β. Here (Xt)t≧0 takes values in some topological vector spaceE and (Wt)t≧0 is anE-valued Brownian motion. We give applications in detail to (infinite volume) quantum fields where β is e.g. a renormalized power of a Schwartz distribution. In addition, we present a new approach to the case of linear β which is based on our general results and second quantization. We also prove new results on general diffusion Dirichlet forms in infinite dimensions, in particular that the Fukushima decomposition holds in this case.

Generalized Mehler semigroups and applications

Let A = (aij ) be a matrix-valued Borel mapping on a domain Ω ⊂ R d , let b = (bi ) be a vector field on Ω, and let LA, b ϕ = a ij ∂ x i ∂ xj ϕ + bi ∂ xi ϕ. We study Borel measures μ on Ω that satisfy the elliptic equation LA, b *μ = 0 in the weak sense: ∫ LA, b ϕ dμ = 0 for all ϕ ∈ C 0 ∞ (Ω). We prove that, under mild conditions, μ has a density. If A is locally uniformly nondegenerate, A ∈ H loc p, 1 and b ∈ L loc p for some p > d, then this density belongs to H loc p, 1. Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.

Strong Solutions of Stochastic Generalized Porous Media Equations: Existence, Uniqueness, and Ergodicity

Abstract We prove a sufficient condition for the closability of classical Dirichlet forms on L 2 ( E ; μ ) which is also necessary if all components of the Dirichlet form are closable. Here E is a locally convex topological vector space and μ, a (not necessarily quasi-invariant) probability measure on E . The same condition is shown to imply the existence of a closed extension whose domain can be described explicitly. In the special case where μ is quasi-invariant with respect to certain vectors in E our result generalises previous theorems on closability. In addition, we prove a Cameron-Martin-type formula for a large class of measures μ. If E is finite dimensional our characterisation of closability is the analogue of the corresponding one-dimensional result. Applications to quantum fields and the connection with the well-studied case of abstract Wiener spaces are discussed.

Existence of strong solutions for stochastic porous media equation under general monotonicity conditions.

Abstract We prove necessary and sufficient conditions on ϑ: R d → R , ϑ ≠ 0 a.e., so that the operator S = Δ + 2ϑ−1▽ϑ · ▽, W1,n, has exactly one self-adjoint extension on W1,p which generates a (sub-)Markovian semigroup on W1,n. These conditions are shown to be always fulfilled if d = 1 and are verified for a large class of functions ϑ if d > 1. We also prove an infinite dimensional analogue of this result where R d is replaced by some topological vector space E and the Lebesgue measure dx by a Gaussian measure μ. The role of Δ is taken by some Ornstein-Uhlenbeck operator with general linear drift and Dom(S) = bounded smooth cylinder functions. Both results are derived as special cases from a general “perturbation” result. We also give the corresponding probabilistic version; i.e., we prove uniqueness for the associated martingale problem. We present in some detail applications to the case where ϑ is a ground state of a Hamiltonian in quantum field theory settling a problem which has been open for some time. Furthermore, we apply our results to prove uniqueness (in distribution) of the diffusion first constructed by G. Jona-Lasinio and P. K. Mitter for the stochastic quantization of F (Φ)2-field theory.

Stochastic Porous Media Equations and Self-Organized Criticality

Abstract Harnack inequalities are established for a class of generalized Mehler semigroups, which in particular imply upper bound estimates for the transition density. Moreover, Poincare and log-Sobolev inequalities are proved in terms of estimates for the square field operators. Furthermore, under a condition, well-known in the Gaussian case, we prove that generalized Mehler semigroups are strong Feller. The results are illustrated by concrete examples. In particular, we show that a generalized Mehler semigroup with an α-stable part is not hyperbounded but exponentially ergodic, and that the log-Sobolev constant obtained by our method in the special Gaussian case can be sharper than the one following from the usual curvature condition. Moreover, a Harnack inequality is established for the generalized Mehler semigroup associated with the Dirichlet heat semigroup on (0,1). We also prove that this semigroup is not hyperbounded.