Homogenization of symmetric Dirichlet forms (original) (raw)

On Instability of Global Path Properties of Symmetric Dirichlet Forms Under Mosco-Convergence

We give sufficient conditions for Mosco convergences for the following three cases: symmetric locally uniformly elliptic diffusions, symmetric Lévy processes, and symmetric jump processes in terms of the L 1 (R d ; dx)-local convergence of the (elliptic) coefficients, the characteristic exponents and the jump density functions, respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conservativeness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.

Weak convergence of symmetric diffusion processes

Probability Theory and Related Fields, 1997

In this paper, we show the convergence of forms in the sense of Mosco associated with the part form on relatively compact open set of Dirichlet forms with locally uniform ellipticity and the locally uniform boundedness of ground states under regular Dirichlet space setting. We also get the same assertion under Dirichlet space in in®nite dimensional setting. As a result of this, we get the weak convergence under some conditions on initial distributions and the growth order of the volume of the balls de®ned by (modi®ed) pseudo metric used in K. Th. Sturm.

On Multidimensional Diffusion Processes with Jumps

Osaka Journal of Mathematics, 2014

Let G be an open set of d (d 2) and d x denotes the Lebesgue measure on it. We construct a diffusion process with jumps associated with diffusion data (diffusion coefficients {ai j (x)}, a drift coefficient {bi (x)} and a killing function c(x)) and a Levy kernel k(x, y) in terms of a lower bounded semi-Dirichlet form on L 2 (G d x). When G is the whole space, we allow that the diffusion coefficients m ay degenerate. We also show some Sobolev inequalities for the Dirichlet form and then show the absolute continuity of its resolvent.

On the Convergence of Dirichlet Processes

Bernoulli, 1999

For a given weakly convergent sequence fX n g of Dirichlet processes we show weak convergence of the sequence of the corresponding quadratic variation processes as well as stochastic integrals driven by the X n values provided that the condition UTD (a counterpart to the condition UT for Dirichlet processes) holds true. Moreover, we show that under UTD the limit process of fX n g is a Dirichlet process, too.

Weak Dirichlet processes with jumps

Stochastic Processes and their Applications, 2017

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process A such that [N, A] = 0, for any continuous local martingale N. Given a function u : [0, T ] × R → R, which is of class C 0,1 (or sometimes less), we provide a chain rule type expansion for u(t, X t) which stands in applications for a chain Itô type rule.

Conservation property of symmetric jump processes

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2011

Motivated by the recent development in the theory of jump processes, we investigate its conservation property. We will show that a jump process is conservative under certain conditions for the volume-growth of the underlying space and the jump rate of the process. We will also present examples of jump processes which satisfy these conditions. Résumé. Motivés par les récents développements dans la théorie des processus de sauts, nous étudions leur propriété de conservation. Nous montrons qu'un processus de saut est conservatif sous certaines conditions sur la croissance du volume de l'espace sous-tendant et sur le taux de saut du processus. Nous donnons des examples de processus satisfaisant ces conditions.

Homogenization of Symmetric Lévy Processes on R d

2018

In this short note we study homogenization of symmetric d-dimensional Lévy processes. Homogenization of one-dimensional pure jump Markov processes has been investigated by Tanaka et al. in [4]; their motivation was the work by Benssousan et al. [1] on the homogenization of diffusion processes in Rd, see also [2] and [10]. We investigate a similar problem for a class of symmetric pure-jump Lévy processes on R d and we identify – using Mosco convergence – the limit process. A symmetric Lévy process (Xt)t≥0 is a stochastic process in R d with stationary and independent increments, càdlàg paths and symmetric laws Xt ∼ −Xt. We can characterize the (finite-dimensional distributions of the) process by its characteristic function Eet, ξ ∈ R, t > 0, which is of the form exp(−tψ(ξ)); due to the symmetry of Xt, the characteristic exponent ψ is real-valued. It is given by the Lévy–Khintchine formula ψ(ξ) = 1 2 〈ξ,Σξ〉+ ∫ h 6=0 ( 1− cos〈ξ, h〉 ) ν(dh), ξ ∈ R. (1) Σ ∈ R is the positive semidefin...

The dirichlet form of a gradient-type drift transformation of a symmetric diffusion

Acta Mathematica Sinica, English Series, 2008

In the context of a symmetric diffusion process X admitting a carré du champs operator, we give a precise description of the Dirichlet form of the process obtained by subjecting X to a drift transformation of gradient type. This description relies on boundary-type conditions restricting an associated reflecting Dirichlet form.

A Dynamic Model for the Two-Parameter Dirichlet Process

Potential Analysis, 2018

Let α = 1/2, θ > −1/2, and ν 0 be a probability measure on a type space S. In this paper, we investigate the stochastic dynamic model for the two-parameter Dirichlet process Π α,θ,ν 0. If S = N, we show that the bilinear form E(F, G) = 1 2 P 1 (N) ∇F (µ), ∇G(µ) µ Π α,θ,ν 0 (dµ), F, G ∈ F , F = {F (µ) = f (µ(1),. .. , µ(d)) : f ∈ C ∞ (R d), d ≥ 1} is closable on L 2 (P 1 (N); Π α,θ,ν 0) and its closure (E, D(E)) is a quasi-regular Dirichlet form. Hence (E, D(E)) is associated with a diffusion process in P 1 (N) which is time-reversible with the stationary distribution Π α,θ,ν 0. If S is a general locally compact, separable metric space, we discuss properties of the model E(F, G) = 1 2 P 1 (S) ∇F (µ), ∇G(µ) µ Π α,θ,ν 0 (dµ), F, G ∈ F , F = {F (µ) = f (φ 1 , µ ,. .. , φ d , µ) : φ i ∈ B b (S), 1 ≤ i ≤ d, f ∈ C ∞ (R d), d ≥ 1}. In particular, we prove the Mosco convergence of its projection forms.

On the conservativeness and the recurrence of symmetric jump-diffusions

Journal of Functional Analysis, 2012

Sufficient conditions for a symmetric jump-diffusion process to be conservative and recurrent are given in terms of the volume of the state space and the jump kernel of the process. A number of examples are presented to illustrate the optimality of these conditions; in particular, the situation is allowed to be that the state space is topologically disconnected but the particles can jump from a connected component to the other components.