Non-symmetric Dirichlet Forms Research Articles

An Averaging Principle for Stochastic Flows and Convergence of Non-Symmetric Dirichlet Forms

We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we prove that, as the deterministic flow is accelerated, the diffusion process converges in law to a diffusion defined on a different space. This averaging principle also holds at the level of the flows. Our contributions in this article include: a proof of an original averaging principle for stochastic flows of kernels; the definition and study of a convergence of sequences of non-symmetric bilinear forms defined on different spaces; the study of weighted Sobolev spaces on metric graphs or “books”.

Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré 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.

Analyticity of Nonsymmetric Ornstein-Uhlenbeck Semigroup with Respect to a Weighted Gaussian Measure

In this paper we show that the realization in $\mathrm {L}^{p}(X,\nu _{\infty })$ of a nonsymmetric Ornstein-Uhlenbeck operator Lp is sectorial for any $p\in (1,+\infty )$ and we provide an explicit sector of analyticity. Here, $(X,\mu _{\infty },H_{\infty })$ is an abstract Wiener space, i.e., X is a separable Banach space, $\mu _{\infty }$ is a centred nondegenerate Gaussian measure on X and $H_{\infty }$ is the associated Cameron-Martin space. Further, $\nu _{\infty }$ is a weighted Gaussian measure, that is, $\nu _{\infty }=e^{-U}\mu _{\infty }$ where U is a convex function which satisfies some minimal conditions. Our results strongly rely on the theory of nonsymmetric Dirichlet forms and on the divergence form of the realization of L2 in $\mathrm {L}^{2}(X,\nu _{\infty })$ .

Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents

We analyze the transience, recurrence, and irreducibility properties of general sub-Markovian resolvents of kernels and their duals, with respect to a fixed sub-invariant measure m. We give a unifying characterization of the invariant functions, revealing the fact that an Lp-integrable function is harmonic if and only if it is harmonic with respect to the weak dual resolvent. Our approach is based on potential theoretical techniques for resolvents in weak duality. We prove the equivalence between the m-irreducible recurrence of the resolvent and the extremality of m in the set of all invariant measures, and we apply this result to the extremality of Gibbs states. We also show that our results can be applied to non-symmetric Dirichlet forms, in general and in concrete situations. A second application is the extension of the so called Fukushima ergodic theorem for symmetric Dirichlet forms to the case of sub-Markovian resolvents of kernels.

Girsanov transformation for non-symmetric Markov processes

In this paper, we study the transformation of an arbitrary non-symmetric Markov process $X$ by multiplicative functionals which are the exponential of continuous additive functionals of $X$ with zero quadratic variations. We characterize the transformed semigroups by their associated quadratic forms. For a pair of dual Markov process $(X,\hat{X})$ associated with a non-symmetric Dirichlet form, we give a necessary and sufficient condition for the Girsanov transformed processes of $X$ and $\hat{X}$ to be dual with respect to another reference measure.

Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and
 non-explosion results

Using elliptic regularity results in weighted spaces, stochastic calculus and the theory of non-symmetric Dirichlet forms, we first showweak existence of non-symmetric distorted Brownian motion for any starting point in some domain $E$ of $\mathbb{R}^d$, where $E$ is explicitly given as the points of strictpositivity of the unique continuous version of the density to its invariant measure. This non-symmetric distorted Brownian motion is also proved to be strong Feller. Non-symmetric distorted Brownian motion is a singular diffusion, i.e. a diffusion that typically hasan unbounded and discontinuous drift. Once having shown weak existence, we obtain from a result of [13] that the constructedweak solution is indeed strong and weakly as well as pathwise unique up to its explosion time. As a consequence of our approach, we can use the theory of Dirichlet formsto prove further properties of the solutions. For example, we obtain new non-explosion criteria for them.We finally present concrete existence and non-explosion results for non-symmetric distorted Brownian motion relatedto a class of Muckenhoupt weights and corresponding divergence free perturbations.

Semilinear elliptic equations with measure data and quasi-regular Dirichlet forms

We are mainly concerned with equations of the form $-Lu=f(x,u)+\mu$, where $L$ is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, $f$ satisfies the monotonicity condition and mild integrability conditions, and $\mu$ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where $L$ corresponds to a lower bounded semi-Dirichlet form.

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting. Itô’s formula in terms of the extended stochastic integrals is obtained.

On the Generalized Feynman–Kac Transformation for Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form \((\mathcal{E},D(\mathcal{E}))\) on L2(E;m). For \(u\in D(\mathcal{E})\), we have Fukushima’s decomposition: \(\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}\). In this paper, we investigate the strong continuity of the generalized Feynman–Kac semigroup defined by \(P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]\). Let \(Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)\) for \(f,g\in D(\mathcal{E})_{b}\). Denote by J1 the dissymmetric part of the jumping measure J of \((\mathcal{E},D(\mathcal{E}))\). Under the assumption that J1 is finite, we show that \((Q^{u},D(\mathcal{E})_{b})\) is lower semi-bounded if and only if there exists a constant α0≥0 such that \(\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}\) for every t>0. If one of these conditions holds, then \((P^{u}_{t})_{t\geq0}\) is strongly continuous on L2(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J1 is finite.

On notions of harmonicity for non-symmetric Dirichlet form

In this paper, we extend the equivalence of the analytic and probabilistic notions of harmonicity in the context of Hunt processes associated with non-symmetric Dirichlet forms on locally compact separable metric spaces. Extensions to the processes associated with semi-Dirichlet forms and nearly symmetric right processes on Lusin spaces including infinite dimensional spaces are mentioned at the end of this paper.

On Representations of Non-Symmetric Dirichlet Forms

In this paper, we present some structure results on non-symmetric Dirichlet forms. These include, in particular, an analogue of LeJan’s transformation rule for their diffusion parts and a Levy-Khintchine type formula for regular non-symmetric Dirichlet forms on Rd.

A note on perturbation of non-symmetric Dirichlet forms by signed smooth measures

This article discusses the perturbation of a non-symmetric Dirichlet form, ( ɛ , D ( ɛ ) ) , by a signed smooth measure μ, where μ = μ1 - μ2 with μ1 and μ2 being smooth measures. It gives a sufficient condition for the perturbed form ( ɛ μ , D ( ɛ μ ) ) (for some α 0 ≥ 0) to be a coercive closed form.

TIME-DEPENDENT DIRICHLET FORMS AND RELATED STOCHASTIC CALCULUS

Analytic and probabilistic properties of symmetric or non-symmetric Dirichlet forms are well studied. But the processes with parabolic generators are out of the framework of symmetric Dirichlet forms. To cover these cases, we have introduced the time-dependent Dirichlet forms and studied their properties so far. In this expository article, we intend to explain in detail the analytic and probabilistic properties for time-dependent Dirichlet forms parallel to the symmetric Dirichlet forms. New results on a characterization of the minimal α-excessive function dominating a quasi-continuous function as well as the correspondence between additive functionals and smooth mesures are given. In particular, we emphasized the existence of the nontrivial semipolar sets under our settings.

Beurling–Deny formula of semi-Dirichlet forms

In this Note we announce a structure result for non-symmetric Dirichlet forms and semi-Dirichlet forms. Our result is regarded as an extension of the celebrated Beurling–Deny formula which is up to now available only for symmetric Dirichlet forms. The result can also be regarded as an extension of Lévy–Khinchine formula or more generally, an extension of Courrège's Theorem in the semi-Dirichlet forms setting. To cite this article: Z.-C. Hu, Z.-M. Ma, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Limits of relaxed dirichlet problems involving a non symmetric dirichlet form

In this paper we study the convergence of solutions of a sequence of relaxed Dirichlet problems relative to non-symmetric Dirichlet forms. The techniques rely on the study of the behaviour of the solutions of the adjoint problems, as suggested by G. Dal Maso and A. Garroni in [16] in the case of linear elliptic operators of second order with bounded measurable coefficients. In particular we prove a compactness results due to Mosco [31] in the symmetric case.

Fractional Derivatives Non-Symmetric and Time-Dependent Dirichlet Forms and the Drift Form

Using fractional derivatives we show that the drift form \int^{\infty}_{–\infty} u(x) \frac{dv(x)}{dx}dx can be approximated by non-symmetric Dirichlet forms. A similar result holds for the drift form in \mathbb R^n with variable coefficients if the coefficient functions satisfy certain regularity and commutator conditions. Since time-dependent Dirichlet forms (in the sense of Y. Oshima) can be interpreted as sums of a drift form (in \tau -direction) and a mixture of \tau -parametrized Dirichlet forms over \mathbb R^n , our results show that time-dependent Dirichlet forms arise as limits of ordinary non-symmetric Dirichlet forms in \mathbb R \times \mathbb R^n -space. An abstract result on fractional powers of Markov generators allows to extend this observation to generalized Dirichlet forms. Another consequence is that the bilinear form induced by an arbitrary Lévy process is the limit of non-symmetric Dirichlet forms.

Representation Formulas for Non-Symmetric Dirichlet Forms

As well-known, for X a given locally compact separable Hausdorff space, m a positive Radon measure on X with supp[ m ] = X and C_0(X) the space of all continuous functions with compact support on X the Beurling and Deny formula states that any regular Dirichlet form (\bar{\mathcal E}, D(\bar{\mathcal E})) on L^2(X,m) can be expressed as \bar{\mathcal E} (u, v) = \bar{\mathcal E}^c (u,v) + \int_X uvk(dx) + \iint_{XxX–d} (u(x) - u(y))((v(x) - v(y))j(dx,dy) for all u,v \in D(\bar{\mathcal E} \cap C_0 (X) where the symmetric Dirichiet form \bar{\mathcal E}^c , the symmetric measure j(dx,dy) and the measure k(dx) are uniquely determined by \bar{\mathcal E} . It is our aim to prove this formula in the non-symmetric case. For this we consider certain families of non-symmetric Dirichlet forms of diffusion type and show that these forms admit an integral representation involving a measure that enjoys some important functional properties as well as in the symmetric case.

Killing and subordination

Given the one-to-one correspondence between nearly Borel right processes and non-symmetric Dirichlet forms, we prove in the present paper that the killing transform of Markov processes is equivalent to strong subordination of the respective Dirichlet forms and give a characterization of so-called bivariate smooth measures.

Markov processes and nonsymmetric Dirichlet forms without regularity

We study right Markov processes in weak duality satisfying the sector condition of Silverstein. Dirichlet space techniques are employed, but without the customary C 0 regularity hypothesis. Nonetheless the potential theory of such processes is quite amenable. In particular, it is proved that semipolar sets are quasi-polar; this extends results of Silverstein and LeJan.

Stochastic calculus related to nonsymmetric Dirichlet forms

On etend des resultats de Fukushima (1980), Fukushima et Takeda (1984) et Takeda, au cas non symetrique et on elargit ainsi l'etendue des applications de la theorie des espaces de Dirichlet