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.
MSC: 35K05 22E25Keywords: Heat kernel Grushin operator Complex Hamilton-Jacobi method Action function Volume form Higher order oscillator a b s t r a c t The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a ''step 2'' Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton-Jacobi method invented by Beals-Gaveau-Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.
In the original publication, the name of the second author has been published incorrectly. The correct name should be "Yasuyuki Oka". The original article has been updated accordingly.
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 nonlocality. 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.
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