General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces (S, B(S), μ), with S Fréchet spaces such that S ⊂ R N , B(S) is the Borel σ -field of S, and μ is a Borel probability measure on S, are introduced. Firstly, a family of non-local Markovian symmetric forms E (α) , 0 < α < 2, acting in each given L 2 (S; μ) is defined, the index α characterizing the order of the non-locality. Then, it is shown that all the forms E (α) defined on n∈N C ∞ 0 (R n ) are closable in L 2 (S; μ). Moreover, sufficient conditions under which the closure of the closable forms, that are Dirichlet forms, become strictly quasi-regular, are given. Finally, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given. The application of the above theorems to the problem of stochastic quantizations of Euclidean Φ 4 d fields, for d = 2, 3, by means of these Hunt processes is indicated.

We consider the quasi-linear Keller-Segel system of singular type, where the principal part u m represents a fast diffusion like 0 < m < 1. We first construct a global weak solution with small initial data in the scaling invariant norm L N(q−m) 2 for all dimensions N 2 and all exponents q 2. As for the large initial data, we show that there exists a blow-up solution in the case of N = 2. In the second part, the decay property in L r with 1 < r < ∞ for 1 − 2 N m < 1 with the mass conservation is shown. On the other hand, in the case of 0 < m < 1 − 2 N , the extinction phenomenon of solution is proved. It is clarified that the case of m = 1 − 2 N exhibits the borderline in the sense that the decay and extinction occur when the diffusion power m changes across 1 − 2 N . For the borderline case of m = 1 − 2 N , our solution decays in L r exponentially as t → ∞.

We consider the degenerate Keller-Segel system (KS) m below. We find the functional space L s (0, T ; L p (R N )) with some p, s for the uniqueness and continuity of weak solutions with respect to the initial data. Our space is discussed from a viewpoint of the scaling invariant class associated with (KS) m for γ = 0. The technique is based on the L 1 -contraction principle for the porous medium equation.

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