Infinite Dimensional Diffusion Processes Research Articles

Stochastic quantization associated with the exp (Phi )_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 (Hoshino et al., J Evol Equ 21:339–375, 2021) on the stochastic quantization of the exp (Phi )_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” vert alpha vert <sqrt{8pi } of the charge parameter alpha . We also identify the solution with an infinite-dimensional diffusion process constructed by the Dirichlet form approach.

Strong uniqueness for Dirichlet operators related to stochastic quantization under exponential/trigonometric interactions on the two-dimensional torus

We consider space-time quantum fields with exponential/trigonometric interactions. In the context of Euclidean quantum field theory, the former and the latter are called the Hoegh-Krohn model and the Sine-Gordon model, respectively. The main objective of the present paper is to construct infinite dimensional diffusion processes which solve modified stochastic quantization equations for these quantum fields on the two-dimensional torus by the Dirichlet form approach and to prove strong uniqueness of the corresponding Dirichlet operators.

Stochastic quantization associated with the $$\exp (\Phi )_2$$-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 $$\exp (\Phi )_{2}$$ -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 it with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an $N$-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of $0.574$. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler–Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

Laguerre and Meixner symmetric functions, and infinite-dimensional diffusion processes

The Laguerre symmetric functions introduced in the note are indexed by arbitrary partitions and depend on two continuous parameters. The top degree homogeneous component of every Laguerre symmetric function coincides with the Schur function with the same index. Thus, the Laguerre symmetric functions form a two-parameter family of inhomogeneous bases in the algebra of symmetric functions. These new symmetric functions are obtained from the N-variate symmetric polynomials of the same name by a procedure of analytic continuation. The Laguerre symmetric functions are eigenvectors of a second-order differential operator, which depends on the same two parameters and serves as the infinitesimal generator of an infinite-dimensional diffusion process X(t). The process X(t) admits approximation by some jump processes related to one more new family of symmetric functions, the Meixner symmetric functions. In equilibrium, the process X(t) can be interpreted as a time-dependent point process on the punctured real line $\mathbb {R}$ {0}, and the point configurations may be interpreted as doubly infinite collections of particles of two opposite charges and with log-gas-type interaction. The dynamical correlation functions of the equilibrium process have determinantal form: they are given by minors of the so-called extended Whittaker kernel, introduced earlier in a paper by Borodin and the author. Bibliography: 28 titles.

A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := {x ∈ [0, 1]N: ∑i≥1xi = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

A Class of Infinite-Dimensional Diffusion Processes with Connection to Population Genetics

Starting from a sequence of independent Wright-Fisher diffusion processes on [0, 1], we construct a class of reversible infinite-dimensional diffusion processes on Δ∞ := { x ∈ [0, 1] N : ∑ i≥1 x i = 1} with GEM distribution as the reversible measure. Log-Sobolev inequalities are established for these diffusions, which lead to the exponential convergence of the corresponding reversible measures in the entropy. Extensions are made to a class of measure-valued processes over an abstract space S. This provides a reasonable alternative to the Fleming-Viot process, which does not satisfy the log-Sobolev inequality when S is infinite as observed by Stannat (2000).

Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices

A homogenization problem for infinite dimensional diffusion processes indexed by Z d having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for the infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an L 1 type homogenization property of the processes with respect to an invariant measure is proved. This is the, so far, best possible analogue in infinite dimensions to a known result in the finite dimensional case (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]). To cite this article: S. Albeverio et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

An elementary proof of the uniqueness of invariant product measures for some infinite dimensional processes

Consider an infinite dimensional diffusion process with state space T Z d , where T is the circle, and defined by an infinitesimal generator L which acts on local functions f as Lf(η)=∑ i∈ Z d ( a i 2(η i) 2 ∂ 2f ∂η i 2 +b i(η) ∂f ∂η i ) . Suppose that the coefficients a i and b i are smooth, bounded, of finite range, have uniformly bounded second order partial derivatives, that a i are uniformly bounded from below by some strictly positive constant, and that a i is a function only of η i . Suppose that there is a product measure ν which is invariant. Then if ν is the Lebesgue measure or if d=1,2, it is the unique invariant measure. Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs are elementary. Similar results can be proved in the context of an interacting particle system with state space {0,1} Z d , with uniformly positive bounded flip rates which are finite range. To cite this article: A.F. Ramı́rez, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 139–144

Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems

Consider an infinite dimensional diffusion process process on T Z d , where T is the circle, defined by the action of its generator L on C2 (T Z d ) local functions as . Assume that the coefficients, a i and b i are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that a i is only a function of and that . Suppose ν is an invariant product measure. Then, if ν is the Lebesgue measure or if d=1,2 , it is the unique invariant measure. Furthermore, if ν is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space {0,1}Z d , defined by the action of its generator on local functions f by , where is the configuration obtained from η altering only the coordinate at site x . Assume that are of finite range, bounded and that . Then, if ν is an invariant product measure for this process, ν is unique when d=1,2 . Furthermore, if ν is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.

FINITE DIMENSIONAL APPROXIMATIONS OF DIFFUSION PROCESSES ON BANACH SPACES

In this paper, we prove that the laws of diffusion processes on a Banach space X associated with quadratic forms (not necessarily symmetric) are the weak limits of laws of finite dimensional diffusions. A new existence proof for the infinite dimensional diffusion processes is obtained as a by-product.

From metropolis to diffusions: Gibbs states and optimal scaling

This paper investigates the behaviour of the random walk Metropolis algorithm in high-dimensional problems. Here we concentrate on the case where the components in the target density is a spatially homogeneous Gibbs distribution with finite range. The performance of the algorithm is strongly linked to the presence or absence of phase transition for the Gibbs distribution; the convergence time being approximately linear in dimension for problems where phase transition is not present. Related to this, there is an optimal way to scale the variance of the proposal distribution in order to maximise the speed of convergence of the algorithm. This turns out to involve scaling the variance of the proposal as the reciprocal of dimension (at least in the phase transition-free case). Moreover, the actual optimal scaling can be characterised in terms of the overall acceptance rate of the algorithm, the maximising value being 0.234, the value as predicted by studies on simpler classes of target density. The results are proved in the framework of a weak convergence result, which shows that the algorithm actually behaves like an infinite-dimensional diffusion process in high dimensions.

Infinite dimensional diffusion processes with singular interaction

An infinite system of hard spheres in R d undergoing Brownian motions and submitted to a smooth pair potential is studied. It is modelized by an infinite-dimensional Stochastic Differential Equation with a local time term. We prove existence and uniqueness of such a diffusion process, and also that Gibbs states are reversible measures.

On the Spectrum of the Generator of an Infinite System of Interacting Diffusions

We study the spectrum of the operator generating an infinite-dimensional diffusion process Ξ (t), in space . Here ν is a “natural”Ξ (t)-invariant measure on which is a Gibbs distribution corresponding to a (formal) Hamiltonian H of an anharmonic crystal, with a value of the inverse temperature β > 0. For β small enough, we establish the existence of an L-invariant subspace such that has a distinctive character related to a “quasi-particle” picture. In particular, has a Lebesgue spectrum separated from the rest of the spectrum of L and concentrated near a point κ1>0 giving the smallest non-zero eigenvalue of a limiting problem associated with β= 0. An immediate corollary of our result is an exponentially fast L 2-convergence to equilibrium for the process Ξ(t) for small values of β.

A finite-dimensional filtering of infinite-dimensional diffusion processes

i. We study the problem of the optimal estimation of an infinite-dimensional diffusion process, given a finite-dimensional vector of observation. For finite-dimensional diffusion processes, an analogous filtering problem can be solved with the help of the known KalmanBusy equations [i]. We derive equations for the covariance function and the best leastsquare estimator of an infinite-dimensional diffusion process, described by a stochastic parabolic partial differential equation. The obtained equations are natural analogs of Kalman-Busy equations and are applicable in the filtering problem for infinite-dimensional diffusion processes.

DIRICHLET FORMS IN TERMS OF WHITE NOISE ANALYSIS II: CLOSABILITY AND DIFFUSION PROCESSES

It is shown that infinite dimensional Dirichlet forms as previously constructed in terms of (generalized) white noise functionals fit into the general framework of classical Dirichlet forms on topological vector spaces. This entails that all results obtained there are applicable. Admissible functionals give rise to infinite dimensional diffusion processes.

Infinite-dimensional diffusion processes as gibbs measures on $$C[0,1]^{Z^d }$$

An infinite lattice system of interacting diffusion processes is characterized as a Gibbs distribution on $$C[0,1]^{Z^d }$$ with continuous local conditional probabilities. Using estimates for the Vasserstein metric onC[0, 1], Dobrushin's contraction technique is applied in order to obtain information about macroscopic properties of the entire diffusion process.

A certain class of infinite dimensional diffusion processes arising in population genetics

A certain class of infinite dimensional diffusion processes arising in population genetics

Non-linear smoothing of infinite-dimensional diffusion processes

In this paper we characterize the conditional law of the ith coordinate of an infinite-dimensional diffusion process with respect to the others If the interaction is given by a smooth gradient system of finite range, the conditional probability is determined in a robust form as the law of a stochastic differential equation with smooth and bounded drift and initial measure. Additionally the conditional law is shown to be Lipschitz continuous in with respect to the Vasserstein metric on C[iX 1]

A certain class of infinite dimensional diffusion process arising in population genetics : Tokuzo Shiga, Tokyo Institute of Technology, Japan

A certain class of infinite dimensional diffusion process arising in population genetics : Tokuzo Shiga, Tokyo Institute of Technology, Japan