Symmetric Dirichlet Forms Research Articles

Essential Spectrum and Feller Type Properties

We give necessary and sufficient conditions for a regular semi-Dirichlet form to enjoy a new Feller type property, which we call weak Feller property. Our characterization involves potential theoretic as well as probabilistic aspects and seems to be new even in the symmetric case. As a consequence, in the symmetric case, we obtain a new variant of a decomposition principle of the essential spectrum for (the self-adjoint operators induced by) regular symmetric Dirichlet forms and a Persson type theorem, which applies e.g. to Cheeger forms on mathsf {RCD^*} spaces.

An elementary proof that walk dimension is greater than two for Brownian motion on Sierpiński carpets

We give an elementary self-contained proof of the fact that the walk dimension of the Brownian motion on an arbitrary generalized Sierpiński carpet is greater than 2, no proof of which in this generality had previously been available in the literature. Our proof is based solely on the self-similarity and hypercubic symmetry of the associated Dirichlet form and on several very basic pieces of functional analysis and the theory of regular symmetric Dirichlet forms. We also present an application of this fact to the singularity of the energy measures with respect to the canonical self-similar measure (uniform distribution) in this case, proved first by M. Hino in [Probab. Theory Related Fields 132 (2005), no. 2, 265–290].

Homogenization of symmetric Dirichlet forms

We consider a homogenization problem for symmetric jump-diffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.

Irreducible decomposition for Markov processes

We prove an irreducible decomposition for Markov processes associated with quasi-regular symmetric Dirichlet forms or local semi-Dirichlet forms under the absolute continuity condition of transition probability with respect to the underlying measure. We do not assume the conservativeness nor the existence of invariant measures for the processes. As applications, we establish a concrete expression for Chacon–Ornstein type ratio ergodic theorem for such Markov processes and show a compactness of semi-groups under the Green-tightness of measures in the framework of symmetric resolvent strong Feller processes without irreducibility.

Heat kernel upper bounds for symmetric Markov semigroups

It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of [6] for Nash-type inequalities with power order considerably and also extend that of [26] for second order differential operators on a complete non-compact manifold.

Decomposition Formulae for Dirichlet Forms and Their Corollaries

We provide decompositions of symmetric Dirichlet forms into recurrent and transient parts as well as into conservative and dissipative parts, in the framework of $$\sigma $$ -finite measure spaces. Combining both formulae, we write every Dirichlet form as the sum of a recurrent, dissipative, and transient-conservative Dirichlet forms. Besides, we prove that Mosco convergence preserves invariant sets and that a Dirichlet form shares the same invariants sets with its approximating Dirichlet forms $${\mathcal {E}}^{(t)}$$ and $${\mathcal {E}}^{(\beta )}$$ . Finally, we show the equivalence between conservativeness (resp. dissipativity) of a Dirichlet form and the conservativeness (reps. dissipativity) of $${\mathcal {E}}^{(t)}$$ and $${\mathcal {E}}^{(\beta )}$$ . The elaborated results are enlightened by some examples.

Characterizations of heat kernel estimates for symmetric non-local Dirichlet forms via resistance forms

Motivated by [5], we obtain new equivalent conditions for two-sided heat kernel estimates of symmetric non-local Dirichlet forms in terms of resistance forms. Characterizations for upper bounds of heat kernel estimates as well as near diagonal lower bounds of Dirichlet heat kernel estimates are also established. These results can be seen as a complement of the recent studies on heat kernel estimates and parabolic Harnack inequalities for symmetric non-local Dirichlet forms in [10, 11].

On singularity of energy measures for symmetric diffusions with full off-diagonal heat kernel estimates

We show that, for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger than two (sub-Gaussian estimates) imply the singularity of the energy measures with respect to the symmetric measure, verifying a conjecture by M. T. Barlow in (Contemp. Math. 338 (2003) 11–40). We also prove that in the contrary case of walk dimension two, that is, where full off-diagonal Gaussian estimates of the heat kernel hold, the symmetric measure and the energy measures are mutually absolutely continuous in the sense that a Borel subset of the state space has measure zero for the symmetric measure if and only if it has measure zero for the energy measures of all functions in the domain of the Dirichlet form.

Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M,d,μ):E(f,g)=E(c)(f,g)+∫M×M(f(x)−f(y))(g(x)−g(y))J(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) is a symmetric Radon measure on M×M. Under general volume doubling condition on (M,d,μ) and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp. two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2.

Renormalized solutions of semilinear elliptic equations with general measure data

In the paper we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia’s definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.

On semilinear elliptic equations with diffuse measures

We consider semilinear equation of the form -Lu=f(x,u)+mu , where L is the operator corresponding to a transient symmetric regular Dirichlet form {mathcal {E}}, mu is a diffuse measure with respect to the capacity associated with {mathcal {E}}, and the lower-order perturbing term f(x, u) satisfies the sign condition in u and some weak integrability condition (no growth condition on f(x, u) as a function of u is imposed). We prove the existence of a solution under mild additional assumptions on {mathcal {E}}. We also show that the solution is unique if f is nonincreasing in u.

An Integrated Version of Varadhan\u2019s Asymptotics for Lower-Order Perturbations of Strong Local Dirichlet Forms

The studies of J. A. Ramírez, Hino–Ramírez, and Ariyoshi–Hino showed that an integrated version of Varadhan’s asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.

FUNCTIONAL ANALYTIC (IR-)REGULARITY PROPERTIES OF SABR-TYPE PROCESSES

The stochastic alpha, beta, rho (SABR) model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in the markets, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR, these are rarely satisfied. We examine here regularity properties of the model from this perspective with a view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups and identify suitable Banach and Hilbert spaces for these. Finally, we comment on the short-time and large time asymptotic behavior of SABR-type processes beyond the heat-kernel framework.

Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller-properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach - and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

Conservativeness criteria for generalized Dirichlet forms

We develop sufficient analytic conditions for conservativeness of non-sectorial perturbations of symmetric Dirichlet forms which can be represented through a carré du champ on a locally compact separable metric space. These form an important subclass of generalized Dirichlet forms which were introduced in [21]. In case there exists an associated strong Feller process, the analytic conditions imply conservativeness, i.e. non-explosion of the associated process in the classical probabilistic sense. As an application of our general results on locally compact separable metric state spaces, we consider a generalized Dirichlet form given on a closed or open subset of Rd which is given as a divergence free first order perturbation of a symmetric energy form. Then using volume growth conditions of the carré du champ and the non-sectorial first order part, we derive an explicit criterion for conservativeness. We present several concrete examples which relate our results to previous ones obtained by different authors. In particular, we show that conservativeness can hold for a large variance if the anti-symmetric part of the drift is strong enough to compensate it. This work continues our previous work on transience and recurrence of generalized Dirichlet forms.

Semilinear elliptic equations with Dirichlet operator and singular nonlinearities

In the paper we consider elliptic equations of the form −Au=u−γ⋅μ, where A is the operator associated with a regular symmetric Dirichlet form, μ is a positive nontrivial measure and γ>0. We prove the existence and uniqueness of solutions of such equations as well as some regularity results. We also study stability of solutions with respect to the convergence of measures on the right-hand side of the equation. For this purpose, we introduce some type of functional convergence of smooth measures, which in fact is equivalent to the quasi-uniform convergence of associated potentials.

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: sym- metric locally uniformly elliptic diffusions, symmetric Levy processes, and symmetric jump processes in terms of the L 1 (R d ;dx)-local convergence of the (elliptic) coefficients, the charac- teristic exponents and the jump density functions, respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conserva- tiveness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.

A remark on the uniqueness of Silverstein extensions of symmetric Dirichlet forms

A remark on the uniqueness of Silverstein extensions of symmetric Dirichlet forms

Intrinsic metrics for non-local symmetric Dirichlet forms and applications to spectral theory

We present a study of what may be called an intrinsic metric for a general regular Dirichlet form. For such forms we then prove a Rademacher type theorem. For strongly local forms we show existence of a maximal intrinsic metric (under a weak continuity condition) and for Dirichlet forms with an absolutely continuous jump kernel we characterize intrinsic metrics by bounds on certain integrals. We then turn to applications on spectral theory and provide for (measure perturbation of) general regular Dirichlet forms an Allegretto–Piepenbrink type theorem, which is based on a ground state transform, and a Shnol' type theorem. Our setting includes Laplacian on manifolds, on graphs and α-stable processes.

Stochastic integration with respect to additive functionals of zero quadratic variation

We consider a Markov process $X$ associated to a nonnecessarily symmetric Dirichlet form $\mathcal{E}$. We define a stochastic integral with respect to a class of additive functionals of zero quadratic variation and then we obtain an It\^{o} formula for the process $u(X)$, when $u$ is locally in the domain of $\mathcal{E}$.