semi-Dirichlet Form 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.

Variational formulas for the exit time of Hunt processes generated by semi-Dirichlet forms

Variational formulas for the Laplace transform of the exit time from an open set of a Hunt process generated by a regular lower bounded semi-Dirichlet form are established. While for symmetric Markov processes, variational formulas are derived for the exponential moments of the exit time. As applications, we provide some comparison theorems and quantitative relations of the exponential moments and Poincaré inequalities.

Large Time Behavior of Solutions to Parabolic Equations with Dirichlet Operators and Nonlinear Dependence on Measure Data

We study large time behavior of renormalized solutions of the Cauchy problem for equations of the form ∂tu − Lu + λu = f(x, u) + g(x, u) ⋅ μ, where L is the operator associated with a regular lower bounded semi-Dirichlet form and μ is a nonnegative bounded smooth measure with respect to the capacity determined by . We show that under the monotonicity and some integrability assumptions on f, g as well as some assumptions on the form , u(t, x) → v(x) as t →∞ for quasi-every x, where v is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.

Systems of quasi-variational inequalities related to the switching problem

We prove the existence of weak solution for a system of quasi-variational inequalities related to a switching problem with dynamic driven by operator associated with a semi-Dirichlet form and with measure data. We give a stochastic representation of solutions in terms of solutions of a system of reflected BSDEs with oblique reflection. As a by-product, we prove the existence of an optimal strategy in the switching problem and show regularity of the payoff function.

Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.

On generalized feynman-kac transformation for markov processes associated with semi-dirichlet forms

Suppose that X is a right process which is associated with a semi-Dirichlet form (ɛ, D(ɛ)) on L2(E; m). Let J be the jumping measure of (ɛ, D(ɛ)) satisfying J(E × E − d) < ro. Let u G D(ɛ)b: = D(ɛ) ∩ L∞ (E; m), we have the following Fukushima's decomposition ũ (Xt) − ũ (Xo) = Mu t + Nut. Define Put f (x) = Ex[eNutf (Xt)]. Let Qu(f, g) = ɛ (f, g) + ɛ (u, fg) for f, g ∈ D(ɛ)b. In the first part, under some assumptions we show that (Qu, D(ɛ)b) is lower semi-bounded if and only if there exists a constant α o ≥ 0 such that ||Ptu ||2 ≤ eα ot for every t > 0. If one of these assertions holds, then (Ptu)t≥ o is strongly continuous on L2(E; m). If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J(E × E − d) < ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of Atand give two sufficient conditions for PtAf (x) = Ex [eAtf (Xt)] to be strongly continuous.

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.

Bivariate Revuz Measures and the Feynman-Kac Formula on Semi-Dirichlet Forms

In this paper we shall first establish the theory of bivariate Revuz correspondence of positive additive functionals under a semi-Dirichlet form which is associated with a right Markov process X satisfying the sector condition but without duality. We extend most of the classical results about the bivariate Revuz measures under the duality assumptions to the case of semi-Dirichlet forms. As the main results of this paper, we prove that for any exact multiplicative functional M of X, the subprocess X M of X killed by M also satisfies the sector condition and we then characterize the semi-Dirichlet form associated with X M by using the bivariate Revuz measure, which extends the classical Feynman-Kac formula.

On multidimensional diffusion processes with jumps

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.

Lévy–Khintchine type representation of Dirichlet generators and semi-Dirichlet forms

Abstract Let U be an open set of ℝ n , m be a positive Radon measure on U such that supp [ m ] = U ${{\rm supp}[m]=U}$ , and ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ be a strongly continuous contraction sub-Markovian semigroup on L 2 ( U ; m ) ${L^2(U;m)}$ . We investigate the structure of ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ . (i) Denote respectively by ( A , D ( A ) ) ${(A,D(A))}$ and ( A ^ , D ( A ^ ) ) ${(\hat{A},D(\hat{A}))}$ the generator and the co-generator of ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ . Under the assumption that C 0 ∞ ( U ) ⊂ D ( A ) ∩ D ( A ^ ) ${C^{\infty }_0(U)\subset D(A)\cap D(\hat{A})}$ , we give an explicit Lévy–Khintchine type representation of A on C 0 ∞ ( U ) ${C^{\infty }_0(U)}$ . (ii) If ( P t ) t &gt; 0 ${(P_t)_{t&amp;gt;0}}$ is an analytic semigroup and hence is associated with a semi-Dirichlet form ( ℰ , D ( ℰ ) ) ${({\cal E}, D({\cal E}))}$ , we give an explicit characterization of ℰ on C 0 ∞ ( U ) ${C^{\infty }_0(U)}$ under the assumption that C 0 ∞ ( U ) ⊂ D ( ℰ ) ${C^{\infty }_0(U)\subset D({\cal E})}$ . We also present a LeJan type transformation rule for the diffusion part of regular semi-Dirichlet forms on general state spaces.

Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms

Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k defined on E × E off the diagonal whose anti-symmetric part is assumed to be less singular than the symmetric part, we construct an associated regular lower bounded semi-Dirichlet form η on L2(E; m) producing a Hunt process X0 on E whose jump behaviours are governed by k. For an arbitrary open subset D ⊂ E, we also construct a Hunt process XD,0 on D in an analogous manner. When D is relatively compact, we show that XD,0 is censored in the sense that it admits no killing inside D and killed only when the path approaches to the boundary. When E is a d-dimensional Euclidean space and m is the Lebesgue measure, a typical example of X0 is the stable-like process that will be also identified with the solution of a martingale problem up to an η-polar set of starting points. Approachability to the boundary ∂ D in finite time of its censored process XD,0 on a bounded open subset D will be examined in terms of the polarity of ∂ D for the symmetric stable processes with indices that bound the variable exponent α(x).

On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets

We obtain a criterion for the quasi-regularity of generalized (non-sectorial) Dirichlet forms, which extends the result of P.J. Fitzsimmons on the quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov) process associated to a semi-Dirichlet form, we present sufficient conditions for a second right process to be a standard one, having the same state space. The above mentioned quasi-regularity criterion is then an application. The conditions are expressed in terms of the associated capacities, nests of compacts, polar sets, and quasi-continuity. The second application is on the quasi-regularity of the generalized Dirichlet forms obtained by perturbing a semi-Dirichlet form with kernels.

Nonlinear filtering of semi-Dirichlet processes

Herein, we consider the nonlinear filtering problem for general right continuous Markov processes, which are assumed to be associated with semi-Dirichlet forms. First, we derive the filtering equations in the semi-Dirichlet form setting. Then, we study the uniqueness of solutions of the filtering equations via the Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. Furthermore, we investigate the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.

Girsanov Transformations for Non-Symmetric Diffusions

Abstract.Let X be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form (ℰ, 𝓓 (ℰ)) on L2(E ;m). For u ∈ 𝓓(ℰ)e, the extended Dirichlet space, we investigate some properties of the Girsanov transformed process Y of X . First, let be the dual process of X and Ŷ the Girsanov transformed process of . We give a necessary and sufficient condition for (Y , Ŷ to be in duality with respect to the measure e2um. We also construct a counterexample, which shows that this condition may not be satisfied and hence (Y , Ŷ ) may not be dual processes. Then we present a sufficient condition under which Y is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Quasi-regular Dirichlet Forms and $L^p$ -resolvents on Measurable Spaces

We prove that for any semi-Dirichlet form \({\left({\varepsilon,D{\left(\varepsilon\right)}}\right)}\) on a measurable Lusin space E there exists a Lusin topology with the given \(\sigma\)-algebra as the Borel \(\sigma\)-algebra so that \({\left({\varepsilon,D{\left(\varepsilon\right)}}\right)}\) becomes quasi-regular. However one has to enlarge E by a zero set. More generally a corresponding result for arbitrary \(L^p\)-resolvents is proven.

WEAK DUALITY AND THE DUAL PROCESS FOR A SEMI-DIRICHLET FORM

We show that the right (Markov) process associated with a quasi-regular semi-Dirichlet form has a dual process, proving that the weak duality hypothesis is satisfied.

Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting

The Lévy–Khintchine formula or, more generally, Courrège's theorem characterizes the infinitesimal generator of a Lévy process or a Feller process on R d . For more general Markov processes, the formula that comes closest to such a characterization is the Beurling–Deny formula for symmetric Dirichlet forms. In this paper, we extend these celebrated structure results to include a general right process on a metrizable Lusin space, which is supposed to be associated with a semi-Dirichlet form. We start with decomposing a regular semi-Dirichlet form into the diffusion, jumping and killing parts. Then, we develop a local compactification and an integral representation for quasi-regular semi-Dirichlet forms. Finally, we extend the formulae of Lévy–Khintchine and Beurling–Deny in semi-Dirichlet forms setting through introducing a quasi-compatible metric.

On the Quasi-regularity of Semi-Dirichlet Forms

We prove that if a right Markov process is associated with a semi-Dirichlet form, then the form is necessarily quasi-regular. As applications, we develop the theory of Revuz measures in the semi-Dirichlet context and we show that quasi-regularity is invariant with respect to time change.