Festschrift Masatoshi Fukushima

The following sections are included:

• Preface
• Contents

Over more than five decades Professor Fukushima has made remarkable contributions to mathematics, especially to stochastic analysis and its applications to science, in particular to mathematical physics. For many of his colleagues, students and friends it becomes an excellent opportunity to express their appreciation of himself and his work by celebrating his Sanju, the 80th birthday in the Japanese tradition, with an international conference and by publishing a Festschrift in his honour. Naturally such a Festschrift should contain a scientific account of Professor Fukushima's oeuvre. When making here an attempt, i.e. writing “un essai” in the classical meaning, to present such an account, we must be aware that we are not discussing a completed oeuvre. Professor Fukushima is continuing to make highly original research contributions with deep insights. Obviously it is pre-matured to describe or guess the lasting impact these latest contributions will have. However it is fair to say that some of his earlier works are meanwhile considered as to belong to the “classical canon” of stochastic analysis, and hence we will concentrate more on those contributions. But we will see how some of his latest research indeed connect with his earlier work in a natural way…

The following sections are included:

• Research Papers (English)
• Expository Writing
• Seminar on Probability (in Japanese)
• Monographs and Textbooks

After recalling basic features of the theory of symmetric quasi regular Dirichlet forms we show how by applying it to the stochastic quantization equation, with Gaussian space-time noise, one obtains weak solutions in a large invariant set. Subsequently, we discuss non symmetric quasi regular Dirichlet forms and show in particular by two simple examples in infinite dimensions that infinitesimal invariance, does not imply global invariance. We also present a simple example of non-Markov uniqueness in infinite dimensions.

We show that there exists an ergodic conductance environment such that the weak (annealed) invariance principle holds for the corresponding continuous time random walk but the quenched invariance principle does not hold. In the present paper we give a proof of the full scaling limit for the weak invariance principle, improving the result in an earlier paper where we obtained a subsequential limit.

We discuss the main stages of development of the error calculation since the beginning of XIX-th century by insisting on what prefigures the use of Dirichlet forms and emphasizing the mathematical properties that make the use of Dirichlet forms more relevant and efficient. The purpose of the paper is mainly to clarify the concepts. We also indicate some possible future research.

The fourth moment theorem provides error bounds of the order in the central limit theorem for elements F of Wiener chaos of any order such that 𝔼(F²) = 1. It was proved by Nourdin and Peccati [31] using Stein's method and the Malliavin calculus. It was also proved by Azmoodeh, Campese and Poly [2] using Stein's method and Dirichlet forms. This paper is an exposition on the connections between Stein's method and the Malliavin calculus and between Stein's method and Dirichlet forms, and on how these connections are exploited in proving the fourth moment theorem.

This paper surveys some of our recent progress on Hardy-type inequalities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in terms of the L²-theory. A crucial application of a result by Fukushima and Uemura (2003) is included. In the second section, the non-linear case (a general Hardy-type inequality) is handled with a direct and analytic proof. In the last section, it is illustrated that the basic estimates presented in the first two sections can still be improved considerably.

In this survey paper we summarize some recent progress on functional inequalities for Dirichlet forms of type

In this work, we will be interested in the push forward measure (φ_t)_*γ, where φ_t is defined by the stochastic differential equation

We modify an argument of Stroock to show that an additive function from one Banach space to another, if universally Gaussian measurable, is necessarily continuous and linear.

We study the behavior of conditional risk measures along decreasing σ-fields. Under a condition of consistency, we prove a non-linear extension of backwards martingale convergence. In particular we show the existence of a limiting conditional risk measure with respect to the tail field, we describe its dual representation in terms of a limiting penalty function, and we show that consistency extends to the tail field. Moreover, we clarify the structure of global risk measures which are consistent with the given sequence of conditional risk measures.

Let us suppose that we have a right continuous Markov semigroup on ℝ^d, d ≥ 1, such that its potential kernel is given by convolution with a function G₀ = g(| · |), where g is decreasing, has a mild lower decay property, and a very weak upper decay property at infinity. This captures not only the Brownian semigroup (classical potential theory) and isotropic α-stable semigroups (Riesz potentials), but also more general isotropic Lévy processes, where the characteristic function has a certain lower scaling property, and various geometric stable processes.

There always exists a corresponding Hunt process. A subset A of ℝ^d is called unavoidable, if the process hits A with probability 1, wherever it starts. It is known that, for any locally finite union of pairwise disjoint balls B(z,r_z), z ∈ Z, which is unavoidable, . The converse is proven assuming, in addition, that, for some ℇ > 0, |z − z′| ≥ ℇ|z|(g(|z|)/g(r_z))^1/d, whenever z, z′ ∈ Z, z ≠ z′. It also holds, if the balls are regularly located, that is, if their centers keep some minimal mutual distance, each ball of a certain size intersects Z, and r_z = g(ϕ(|z|)) where ϕ is a decreasing function.

The results generalize and, exploiting a zero-one law, simplify recent work by A. Mimica and Z.Vondraček.

We introduce the concept of functions of locally bounded variation on abstract Wiener spaces and study their properties. Some nontrivial examples and applications to stochastic analysis are also discussed.

Stochastic processes on the field of p-adic numbers have been studied for more than two decades. The theory of Dirichlet space is applied in important parts of the studies. In some recent observations, the hierarchical structure is handled by tree structure and the structure gives an overview for measure symmetric stochastic processes on the field of p-adic numbers and ends of a tree. The existing researches were undertaken by relying on structure of eigenfunctions determined by the structural design of branches at every node. The objective of the present article is focusing on the case where a Dirichlet form is given at every node and tier linkage is given by bilinear forms coming from pairs of nodes connected by branch or stemmed directly from their identical parental node. In other words, we will deal with the case where the structure of eigenfunctions differs from the one in existing studies.

We intend to highlight the impact that the theory of local Dirichlet forms has had on the formulation of dynamics in Schroedinger theory, in stochastic models for large scale natural phenomena, and in the theory of relativistic quantum fields where perturbative methods are bound to fail.

We give a necessary and sufficient condition on the global stability of the fundamental solution for generalized Feynman-Kac semigroup of (relativistic) stable-like processes in terms of the gauge function of Feynman-Kac semigroup under mild conditions for related measures appeared in the perturbations.

In this paper we study the Martin boundary of open sets with respect to a large class of purely discontinuous symmetric Lévy processes in ℝ^d. We show that, if D ⊂ ℝ^d is an open set which is κ-fat at a boundary point Q ∈ ∂D, then there is exactly one Martin boundary point associated with Q and this Martin boundary point is minimal.

We study the level statistics of one-dimensional Schrödinger operator with random potential decaying like x^-α at infinity. We consider the point process ξ_L consisting of the rescaled eigenvalues and show that: (i) (ac spectrum case) for , ξ_L converges to a clock process, and the fluctuation of the eigenvalue spacing converges to Gaussian. (ii) (critical case) for , ξ_L converges to the limit of the circular β-ensemble.

The loop measure is associated with a Markov generator. We compute the variation of the loop measure induced by an infinitesimal variation of the generator affecting the killing rates or the jumping rates.

The regular subspaces of a Dirichlet form are the regular Dirichlet forms that inherit the original form but possess smaller domains. The two problems we are concerned are: (1) the existence of regular subspaces of a fixed Dirichlet form, (2) the characterization of the regular subspaces if exists. In this paper, we will first research the structure of regular subspaces for a fixed Dirichlet form. The main results indicate that the jumping and killing measures of each regular subspace are just equal to that of the original Dirichlet form. By using the independent coupling of Dirichlet forms and some celebrated probabilistic transformations, we will study the existence and characterization of the regular subspaces of local Dirichlet forms.

In this survey paper we review the concept of quasi-regularity and report some of our recent progress on quasi-regular semi-Dirichlet forms. We start with an introduction to the concept of quasi-regularity and then review three results closely related to this concept. Afterwards we report our recent results on representations of semi-Dirichlet forms and the Fukushima type decomposition for Markov processes associated with quasi-regular semi-Dirichlet forms. We report also our results on semi-h-transform of semi-Dirichlet forms, some applications of semi-Dirichlet forms in the study of nonlinear filtering, and our recent investigation concerning Hunt's hypothesis (H).

Motivated by downside risk minimization on the wealth process in an incomplete market model, we consider minimizing the probability that the empirical mean of a controlled semi-martingale falls below a certain level on a time horizon T, and then study the asymptotic behavior of the minimizing probability as T → ∞. This asymptotic behavior relates to an risk-sensitive stochastic control problem in the risk-averse case. Indeed, we can obtain an expression of the decay rate of the probability by the Legendre transform of the limit value of the value function of the stochastic control problem, which is characterized as the solution to an H-J-B equation of ergodic type.

In this paper, we obtain a comparison theorem for backward stochastic partial differential equation (SPDEs) with jumps. We apply it to introduce space-dependent convex risk measures as a model for risk in large systems of interacting components.

In this paper, we use the analytic and the stochastic calculus related to the lower bounded semi-Dirichlet forms to construct a diffusion process associated with a differential generator containing a boundary condition on smooth compact sets. For such differential generator, we give a condition under which the associated bilinear form becomes a semi-Dirichlet form. Once the condition is satisfied, then there corresponds a diffusion process. By using the stochastic calculus related to the semi-Dirichlet forms, we give a representation of the solution of a partial differential equation with boundary and terminal value conditions. Moreover a stochastic representation of the diffusion process is given.

Let k : E × E → [0, ∞) be a non-negative measurable function on some locally compact separable metric space E. We provide some simple conditions such that the quadratic form with jump kernel k becomes a regular lower bounded (non-local, non-symmetric) semi-Dirichlet form. If E = ℝⁿ we identify the generator of the semi-Dirichlet form and its (formal) adjoint. In particular, we obtain a closed expression of the adjoint of the stable-like generator −(−Δ)^α(x) in the sense of Bass. Our results complement a recent paper by Fukushima and Uemura [3] and establish the relation of these results with the symmetric principal value (SPV) approach due to Zhi-ming Ma and co-authors [5].

We discuss the ultracontractivity for non-symmetric Markovian semigroup. To do this, we make use of the symmetric bilinear form. For an application, we deal with the exponential convergence rate of the semigroup. We show that a non-symmetric semigroup converges faster than the symmetric semigroup. We give same examples of diffusions on Riemannian manifolds.

The goal of this paper is twofold: we study metric measure spaces (X, d, m) with variable lower bounds for the Ricci curvature and we study pathwise coupling of Brownian motions. Given any lower semicontinuous function k : X → ℝ we introduce the curvature-dimension condition CD(k, ∞) which canonically extends the curvature-dimension condition CD(K, ∞) of Lott-Sturm-Villani for constant K ∈ R. For infinitesimally Hilbertian spaces we prove

• its equivalence to an evolution-variation inequality EVI_k which in turn extends the EVI_K-inequality of Ambrosio-Gigli-Savaré;
• its stability under convergence and its local-to-global property.

For metric measure spaces with uniform lower curvature bounds K we prove that for each pair of initial distributions µ₁, µ₂ on X there exists a coupling , t ≥ 0, of two Brownian motions on X with the given initial distributions such that a.s.

We consider a diffusion operator associated with a skew product of a one-dimensional generalized diffusion process R and the spherical Brownian motion Θ with respect to a positive continuous additive functional f(t) of R. We give sufficient conditions for [SSP] (i.e., 1 is a semismall perturbation of the operator ) and [IU] (i.e., the associated heat kernel is intrinsically ultracontractive) in terms of characteristics of R and f(t).

The following section is included:

• Author Index