semi-Dirichlet Forms Research Articles

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.

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.

The complement value problem for non-local operators

Let [Formula: see text] be a bounded Lipschitz domain of [Formula: see text]. We consider the complement value problem [Formula: see text] Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in our approach.

Quasimartingales associated to Markov processes

For a fixed right process $X$ we investigate those functions $u$ for which $u(X)$ is a quasimartingale. We prove that $u(X)$ is a quasimartingale if and only if $u$ is the dif- ference of two finite excessive functions. In particular, we show that the quasimartingale nature of $u$ is preserved under killing, time change, or Bochner subordination. The study relies on an analytic reformulation of the quasimartingale property for $u(X)$ in terms of a certain variation of $u$ with respect to the transition function of the process. We provide sufficient conditions under which $u(X)$ is a quasimartingale, and finally, we extend to the case of semi-Dirichlet forms a semimartingale characterization of such functionals for symmetric Markov processes, due to Fukushima.

Stochastic calculus for Markov processes associated with semi-Dirichlet forms

Let $(\mathcal{E},D(\mathcal{E}))$ be a quasi-regular semi-Dirichlet form and $(X_t)_{t\geq0}$ be the associated Markov process. For $u\in D(\mathcal{E})_{loc}$, denote $A_t^{[u]}:=\tilde{u}(X_{t})-\tilde{u}(X_{0})$ and $F^{[u]}_t:=\sum_{0<s\leq t}(\tilde u(X_{s})-\tilde u(X_{s-}))1_{\{|\tilde u(X_{s})-\tilde u(X_{s-})|>1\}}$, where $\tilde{u}$ is a quasi-continuous version of $u$. We show that there exist a unique locally square integrable martingale additive functional $Y^{[u]}$ and a unique continuous local additive functional $Z^{[u]}$ of zero quadratic variation such that $$A_t^{[u]}=Y_t^{[u]}+Z_t^{[u]}+F_t^{[u]}.$$ Further, we define the stochastic integral $\int_0^t\tilde v(X_{s-})dA_s^{[u]}$ for $v\in D(\mathcal{E})_{loc}$ and derive the related It\^{o}'s formula.

On the Conservativeness of Some Markov Processes

We give a unified method to obtain the conservativeness of a class of Markov processes associated with lower bounded semi-Dirichlet forms on L 2(X;m), including symmetric diffusion processes, some non-symmetric diffusion processes and jump type Markov processes on X, where X is a locally compact separable metric space and m is a positive Radon measure on X with full topological support. Using the method, we give an example in each section, providing the conservativeness of the processes, that are given by the “increasingness of the volume of some sets(balls)” and “that of the coefficients on the sets” of the Markov processes.

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.

Fukushima type decomposition for semi-Dirichlet forms

We present a Fukushima type decomposition in the setting of general quasi-regular semi-Dirichlet forms. The decomposition is then employed to give a transformation formula for martingale additive functionals. Applications of the results to some concrete examples of semi-Dirichlet forms are given at the end of the paper. We discuss also the uniqueness question about the Doob-Meyer decomposition on optional sets of interval type.

On Dirichlet Forms and Semi-Dirichlet Forms

One aim of this note is to give an overview of some developments in the area of Dirichlet forms. A second aim is to review the new book “Semi-Dirichlet forms and Markov processes” by Yoichi Oshima. This book was published by de Gruyter (Berlin) in 2013, but first versions were written as lecture notes 25 years ago. We first give a rather short and light introduction into the field of Dirichlet forms with a special emphasis on the subjects presented in the book under consideration. After a small account on the history of Dirichlet forms we comment on the book by Oshima against the background of related works.

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.

Semi-Dirichlet forms, Feynman–Kac functionals and the Cauchy problem for semilinear parabolic equations

In the first part of the paper we prove various results on regularity of Feynman–Kac functionals of Hunt processes associated with time-dependent semi-Dirichlet forms. In the second part we study the Cauchy problem for semilinear parabolic equations with measure data involving operators associated with time-dependent forms. Model examples are non-symmetric divergence form operators and fractional laplacians with possibly variable exponents. We first introduce a definition of a solution resembling Stampacchia's definition in the sense of duality and then, using the results of the first part, we prove the existence, uniqueness and regularity of solutions of the problem under mild assumptions on the data.

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.

FUKUSHIMA'S DECOMPOSITION FOR DIFFUSIONS ASSOCIATED WITH SEMI-DIRICHLET FORMS

Diffusion processes associated with semi-Dirichlet forms are studied in the paper. The main results are Fukushima's decomposition for the diffusions and a transformation formula for the corresponding martingale part of the decomposition. The results are applied to some concrete examples.

Balayage of Semi-Dirichlet Forms

AbstractIn this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayagedmeasures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

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 Calabi’s Strong Maximum Principle via Local Semi-Dirichlet Forms

We give a stochastic proof of an extension of E. Calabi’s strong maximum principle under some geometric conditions in the framework of strong Feller diffusion processes associated to local regular semi-Dirichlet forms with lower bounds. As a corollary, our notion of subharmonicity implies a notion of viscosity subsolution in a stochastic sense. We can apply our result to singular geometric object like Alexandrov space, limit space under spectral distance of Riemannian manifolds with uniform lower Ricci curvature bound and so on.

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.

Invariant sets and ergodic decomposition of local semi-Dirichlet forms

Invariant sets and ergodic decomposition of local semi-Dirichlet forms

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.

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.