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Register a new userNon-local Markovian symmetric forms on infinite dimensional spaces
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General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {cal B}(S), mu)$, with $S$ Fr{e}chet spaces such that $S subset {mathbb R}^{mathbb N}$, ${cal B}(S)$ is the Borel $sigma$-field of $S$, and $mu$ is a Borel probability measure on $S$, are introduced. Firstly, a family of non-local Markovian symmetric forms ${cal E}_{(alpha)}$, $0 < alpha < 2$, acting in each given $L^2(S; mu)$ is defined, the index $alpha$ characterizing the order of the non-locality. Then, it is shown that all the forms ${cal E}_{(alpha)}$ defined on $bigcup_{n in {mathbb N}} C^{infty}_0({mathbb R}^n)$ are closable in $L^2(S;mu)$. 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 $Phi^4_d$ fields, for $d =2, 3$, by means of these Hunt processes is indicated.
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The general framework on the non-local Markovian symmetric forms on weighted $l^p$ $(p in [1, infty])$ spaces constructed by [A,Kagawa,Yahagi,Y 2020], by restricting the situation where $p =2$, is applied to such measure spaces as the space cut-off $P(phi)_2$ Euclidean quantum field, the $2$-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the {it{non-local}} type stochastic quantization is constructed.
In the context of a metric measure Dirichlet space satisfying volume doubling and Poincare inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.
Let $mathcal{M}$ be an atomless semifinite von Neumann algebra (or an atomic von Neumann algebra with all atoms having the same trace) acting on a (not necessarily separable) Hilbert space $H$ equipped with a semifinite faithful normal trace $tau$. Let $E(mathcal{M},tau) $ be a symmetric operator space affiliated with $ mathcal{M} $, whose norm is order continuous and is not proportional to the Hilbertian norm $left|cdotright|_2$ on $L_2(mathcal{M},tau)$. We obtain general description of all bounded hermitian operators on $E(mathcal{M},tau)$. This is the first time that the description of hermitian operators on asymmetric operator space (even for a noncommutative $L_p$-space) is obtained in the setting of general (non-hyperfinite) von Neumann algebras. As an application, we resolve a long-standing open problem concerning the description of isometries raised in the 1980s, which generalizes and unifies numerous earlier results.
This paper establishes results on the existence and uniqueness of solutions to McKean-Vlasov equations, also called mean-field stochastic differential equations, in an infinite-dimensional Hilbert space setting with irregular drift. Here, McKean-Vlasov equations with additive noise are considered where the driving noise is cylindrical (fractional) Brownian motion. The existence and uniqueness of weak solutions are established for drift coefficients that are merely measurable, bounded, and continuous in the law variable. In particular, the drift coefficient is allowed to be singular in the spatial variable. Further, we discuss existence of a pathwisely unique strong solution as well as Malliavin differentiability.
In the present paper, we are aiming to study limiting behavior of infinite dimensional Volterra operators. We introduce two classes $tilde {mathcal{V}}^+$ and $tilde{mathcal{V}}^-$of infinite dimensional Volterra operators. For operators taken from the introduced classes we study their omega limiting sets $omega_V$ and $omega_V^{(w)}$ with respect to $ell^1$-norm and pointwise convergence, respectively. To investigate the relations between these limiting sets, we study linear Lyapunov functions for such kind of Volterra operators. It is proven that if Volterra operator belongs to $tilde {mathcal{V}}^+$, then the sets and $omega_V^{(w)}(xb)$ coincide for every $xbin S$, and moreover, they are non empty. If Volterra operator belongs to $tilde {mathcal{V}}^-$, then $omega_V(xb)$ could be empty, and it implies the non-ergodicity (w.r.t $ell^1$-norm) of $V$, while it is weak ergodic.
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