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Seiichiro Kusuoka

Publications and source records attributed to Seiichiro Kusuoka.

At least 19 recordsLinked to original sources
arXiv2026

Convergence to the Dynamical $Φ^4_3$ Model under a Vanishing Quintic Perturbation I: A Paracontrolled Approach

We study local convergence to the dynamical $Φ^4_3$ model under a vanishing quintic perturbation. More precisely, on the three-dimensional torus we consider $$\partial_tu_\varepsilon=Δu_\varepsilon-\varepsilon^αu_\varepsilon^5+ξ_\varepsilon+C_\varepsilon u_\varepsilon+\widetilde C_\varepsilon u_\varepsilon^3$$ for $α\in(\frac56,1)$, where $ξ_\varepsilon$ is a spatial mollification of space-time white noise. Although the quintic coefficient vanishes, its contractions generate divergent linear and cubic contributions. We identify suitable mass and cubic counterterms that compensate these divergences. Using paracontrolled calculus, we construct the required enhanced stochastic data and prove that their renormalized higher-order components vanish, while the remaining coordinates converge to the enhanced data of the dynamical $Φ^4_3(λ)$ model. We further establish local well-posedness and stability of the associated deterministic solution map. Consequently, for well-prepared initial conditions, $u_\varepsilon$ converges in probability, as a random local solution germ, to the renormalized dynamical $Φ^4_3(λ)$ solution. In particular, the cubic counterterm allows convergence strictly below the threshold $α=1$ arising when only linear renormalization is allowed.

math.PR↗
arXiv2026

Nonequilibrium steady state in Lindblad dynamics for infinite quantum spin systems

We consider Lindblad dynamics of quantum spin systems on infinite lattices and define a nonequilibrium steady state (NESS) and a time-averaged nonequilibrium steady state (TANESS) on the basis of $C^*$-algebraic formalism. Generically, the NESS on an infinite system does not equal the thermodynamic limit of NESSs on finite systems. We give a sufficient condition that they coincide with each other, in terms of both a condition number, which quantifies the normality of a Liouvillian, and some spectral gaps on finite subsystems. To appreciate the importance of the condition number, we provide an example in which the spectral gaps have nonzero lower bounds uniformly for any finite subsystems but a thermodynamic limit and a long-time limit (or a long-time average) do not commute with each other.

math-ph↗
arXiv2026

A quantitative replica-symmetric bound for Sherrington--Kirkpatrick model in the entire de Almeida--Thouless region

We consider the Sherrington--Kirkpatrick model with inverse temperature $β>0$ and deterministic external field $h>0$. Let $q$ be the replica-symmetric fixed point: $q=\mathbb E{\rm tanh}^2 (h+β\sqrt q\,Z)$, where $Z$ is a standard normal. We prove that, uniformly on compact subsets of the strict de Almeida--Thouless region: $β^2\mathbb E{\rm sech}^4(h+β\sqrt{q} Z) <1,$ the overlap satisfies the concentration: $$ \mathbb E\langle (R_{12}-q)^2\rangle=O(N^{-1}). $$ As a consequence, we obtain an $O(N^{-1})$ replica-symmetric free-energy correction and identify the finite-volume replicon susceptibility. Our proof is self-contained and does not use the identification of the limiting free energy with the Parisi variational formula. Moreover, we prove the central limit theorem for the overlap in this region. The present paper provides an alternative proof of the replica-symmetric free energy formula in the de Almeida--Thouless region, which was recently established by Lopatto [arXiv:2604.11921]. An advantage of our approach is that it establishes an explicit quantitative bound and yields the replica-symmetric free energy formula as a consequence. The main result supersedes the corresponding result in our recent preprint arXiv:2607.23427, extending the replica-symmetric bounds to the strict de Almeida-Thouless region. However, we keep the previous preprint, since its argument is different and substantially simpler than the one given here.

math.PR↗
arXiv2026

A note on Lata\la's argument in SK model

In this note, we consider the Sherrington--Kirkpatrick model with deterministic external field. Let $q=q(β,h)$ denote the solution of the replica-symmetric self-consistency equation \[ q=\mathbb E\tanh^2\!\left(h+β\sqrt q\,Z\right), \qquad Z\sim N(0,1), \] where $β$ and $h$ are inverse temperature and external field, respectively. By refining Lata\la' s argument, previously limited to \(β< \frac{1}{2}\), and using the Kearns--Saul inequality, we prove overlap concentration and convergence of the free energy to the replica symmetric formula with error \(O(N^{-1})\) whenever \[ β^2\frac{q}{{\rm arctanh}q}<1. \] Note that for any $β<1$ and $h\in \mathbb R$, the condition above is satisfied. Moreover, for every nonzero $h$, this region contains a nonempty interval with $β>1$.

math.PR↗
arXiv2026

Singularity of solutions to singular SPDEs

Building on the notes [Hai17], we give a sufficient condition for the marginal distribution of the solution of singular SPDEs on the $d$-dimensional torus to be singular with respect to the law of the Gaussian measure induced by the linearised equation. As applications we obtain the singularity of the $Φ^4_3$-measure with respect to the Gaussian free field measure and the border of parameters for the fractional $Φ^4$-measure to be singular with respect to the Gaussian free field measure. Our approach is applicable to quite a large class of singular SPDEs.

math.PR↗
arXiv2025

Stochastic quantization of the weighted exponential QFT

We consider the stochastic quantization equation associated with the weighted exponential quantum field model (or the Høegh-Krohn model) on the two dimensional torus. Unlike in the case of the usual (unweighted) exponential model, the drift term of the stochastic quantization equation can be both positive and negative, and that makes the equation more difficult to treat. We prove the unique existence of the time-global solution under a certain initial condition by a pathwise PDE argument in the so-called $L^2$-regime. We also see that this solution is properly associated with a Dirichlet form canonically constructed from the weighted exponential quantum field measure.

math.PR↗
arXiv2025

Stochastic quantization of the three-dimensional polymer measure via the Dirichlet form method

We prove that there exists a diffusion process whose invariant measure is the three dimensional polymer measure $ν_λ$ for all $λ>0$. We follow in part a previous incomplete unpublished work of the first named author with M. Röckner and X.Y. Zhou. For the construction of $ν_λ$ we rely on previous work by J. Westwater, E. Bolthausen and X.Y. Zhou. Using $ν_λ$, the diffusion is constructed by means of the theory of Dirichlet forms on infinite-dimensional state spaces. The closability of the appropriate pre-Dirichlet form which is of gradient type is proven, by using a general closability result in [AR89a]. This result does not require an integration by parts formula (which does not even hold for the two-dimensional polymer measure $ν_λ$) but requires the quasi-invariance of $ν_λ$ along a basis of vectors in the classical Cameron-Martin space such that the Radon-Nikodym derivatives have versions which form a continuous process.

math.PR↗
arXiv2024

Berry-Esseen bounds for large-time asymptotics of one-dimensional diffusion processes via Malliavin-Stein method

We consider solutions of stochastic differential equations which diverge to infinity as the time parameter goes to infinity. If the coefficients converge as the spacial variable goes to infinity, then the solutions will get close to some Gaussian processes with positive drifts as the time parameter goes to infinity. In this paper, we prove Berry-Esseen type bounds for the solutions in this setting. In particular, we obtain bounds of the total variation distance between the law of the centered and scaled solutions of the stochastic differential equations and the standard normal distribution with an optimal rate of convergence in the time parameter. In the proof we apply the Malliavin-Stein method to estimate the total variation distance.

math.PR↗
arXiv2022

Remarks on Stochastic Systems I: Markov properties, local and global uniqueness, and limits of stochastic equations

In the present paper, we give some examples of stochastic differential equations which have delicateness in the Markov and strong Markov properties, the uniqueness locally in time and globally in time, and initial conditions. Moreover, we show that such stochastic differential equations appear in the limits of stochastic differential equations which have the existence and pathwise uniqueness of solutions. These examples are constructed in motivation to singular stochastic partial differential equations. We also give some examples of shifted equations whose sum of the solutions depends on the choices of the decomposition of the initial condition of the original equation.

math.PR↗
arXiv2022

Construction of a non-Gaussian and rotation-invariant $Φ^4$-measure and associated flow on ${\mathbb R}^3$ through stochastic quantization

A new construction of non-Gaussian, rotation-invariant and reflection positive probability measures $μ$ associated with the $φ^4_3$-model of quantum field theory is presented. Our construction uses a combination of semigroup methods, and methods of stochastic partial differential equations (SPDEs) for finding solutions and stationary measures of the natural stochastic quantization associated with the $φ^4_3$-model. Our starting point is a suitable approximation $μ_{M,N}$ of the measure $μ$ we intend to construct. $μ_{M,N}$ is parametrized by an $M$-dependent space cut-off function $ρ_M: {\mathbb R}^3\rightarrow {\mathbb R}$ and an $N$-dependent momentum cut-off function $ψ_N: \widehat{\mathbb R}^3 \cong {\mathbb R}^3 \rightarrow {\mathbb R}$, that act on the interaction term (nonlinear term and counterterms). The corresponding family of stochastic quantization equations yields solutions $(X_t^{M,N}, t\geq 0)$ that have $μ_{M,N}$ as an invariant probability measure. By a combination of probabilistic and functional analytic methods for singular stochastic differential equations on negative-indices weighted Besov spaces (with rotation invariant weights) we prove the tightness of the family of continuous processes $(X_t^{M,N},t \geq 0)_{M,N}$. Limit points in the sense of convergence in law exist, when both $M$ and $N$ diverge to $+\infty$. The limit processes $(X_t; t\geq 0)$ are continuous on the intersection of suitable Besov spaces and any limit point $μ$ of the $μ_{M,N}$ is a stationary measure of $X$. $μ$ is shown to be a rotation-invariant and non-Gaussian probability measure and we provide results on its support. It is also proven that $μ$ satisfies a further important property belonging to the family of axioms for Euclidean quantum fields, it is namely reflection positive.

math.PR↗
arXiv2022

An improvement of the integrability of the state space of the $Φ^4_3$-process and the support of the $Φ^4_3$-measure constructed by the limit of stationary processes of approximating stochastic quantization equations

We improve the integrability of the state space of the $Φ^4_3$-process and the support of the $Φ^4_3$-measure on the torus obtained in [Albeverio, Kusuoka, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2020]. For the improvement, we improve the estimates of the Hölder continuity in time of the solutions to approximation equations. In the present paper, we only discuss the estimates different from those in [Albeverio, Kusuoka, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 2020].

math.PR↗
arXiv2022

Stochastic quantization associated with the $\exp(Φ)_2$-quantum field model driven by space-time white noise on the torus in the full $L^1$-regime

The present paper is a continuation of our previous work on the stochastic quantization of the $\exp(Φ)_2$-quantum field model on the two-dimensional torus. Making use of key properties of Gaussian multiplicative chaos and refining the method for singular SPDEs introduced in the previous work, we construct a unique time-global solution to the corresponding parabolic stochastic quantization equation in the full "$L^{1}$-regime" $\vertα\vert<\sqrt{8π}$ of the charge parameter $α$. We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.

math.PR↗
arXiv2021

Recurrence of direct products of diffusion processes in random media having zero potentials

In this paper, we introduce an index which measures the strength of recurrence of symmetric Markov processes, and give some sufficient conditions for recurrence of direct products of symmetric diffusion processes. The index is given by the Dirichlet forms of the Markov processes. Moreover, as an application, we prove the recurrence of some multi-dimensional diffusion processes in random environments including zero potentials.

math.PR↗
arXiv2021

The invariant measure and the flow associated to the $Φ^4_3$-quantum field model

We give a direct construction of invariant measures and global flows for the stochastic quantization equation to the quantum field theoretical $Φ^4_3$-model on the $3$-dimensional torus. This stochastic equation belongs to a class of singular stochastic partial differential equations (SPDEs) presently intensively studied, especially after Hairer's groundbreaking work on regularity structures. Our direct construction exhibits invariant measures and flows as limits of the (unique) invariant measures for corresponding finite dimensional approximation equations. Our work is done in the setting of distributional Besov spaces, adapting semigroup techniques for solving nonlinear dissipative parabolic equations on such spaces and using methods that originated from work by Gubinelli et al on paracontrolled distributions for singular SPDEs.

math.PR↗
arXiv2020

Stochastic quantization associated with the $\exp(Φ)_2$-quantum field model driven by space-time white noise on the torus

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $\exp (Φ)_{2}$-quantum field model or Høegh-Krohn's model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation, and identify with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

math.PR↗
arXiv2016

Hölder and Lipschitz continuity of the solutions to parabolic equations of the non-divergence type

We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on all coefficients. Under the assumptions we show the Hölder continuity of the solution in the spatial component. Furthermore, additionally assuming the Dini continuity of the coefficient of the second order derivative, we have the better continuity of the solution. In the proof, we use a probabilistic method, in particular the coupling method. As a corollary, under an additional assumption we obtain the Hölder and Lipschitz continuity of the fundamental solution in the spatial component.

math.AP↗
arXiv2015

Characterization of the convergence in total variation and extension of the Fourth Moment Theorem to invariant measures of diffusions

We give necessary and sufficient conditions to characterize the convergence in distribution of a sequence of arbitrary random variables to a probability distribution which is the invariant measure of a diffusion process. This class of target distributions includes the most known continuous probability distributions. Precisely speaking, we characterize the convergence in total variation to target distributions which are not Gaussian or Gamma distributed, in terms of the Malliavin calculus and of the coefficients of the associated diffusion process. We also prove that, among the distributions whose associated squared diffusion coefficient is a polynomial of second degree (with some restrictions on its coefficients), the only possible limits of sequences of multiple integrals are the Gaussian and the Gamma laws.

math.PR↗