Theory Of Dirichlet Forms Research Articles

Validity and failure of the integral representation of Γ-limits of convex non-local functionals

We prove an integral-representation result for limits of non-local quadratic forms on H01(Ω), with Ω a bounded open subset of Rd, extending the representation on Cc∞(Ω) given by the Beurling-Deny formula in the theory of Dirichlet forms. We give a counterexample showing that a corresponding representation may not hold if we consider analogous functionals in W01,p(Ω), with p≠2 and 1<p⩽d.

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].

Stochastic Ricci Flow on Compact Surfaces

Abstract In this paper we introduce the stochastic Ricci flow (SRF) in two spatial dimensions. The flow is symmetric with respect to a measure induced by Liouville conformal field theory. Using the theory of Dirichlet forms, we construct a weak solution to the associated equation of the area measure on a flat torus, in the full “$L^1$ regime” $\sigma &amp;lt; \sigma _{L^1}=2 \sqrt \pi $ where $\sigma $ is the noise strength. We also describe the main necessary modifications needed for the SRF on general compact surfaces and list some open questions.

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 $\mathrm {L}^{2}(X,\nu _{\infty })$ .

Backward Stochastic Differential Equations and Dirichlet Problems of Semilinear Elliptic Operators with Singular Coefficients

In this paper, we prove that there exists a unique solution to the Dirichlet boundary value problem for a general class of semilinear second order elliptic differential operators which do not necessarily have the maximum principle and are non-symmetric in general. Our method is probabilistic. It turns out that we need to solve a class of backward stochastic differential equations with singular coefficients, which is of independent interest itself. The theory of Dirichlet forms also plays an important role.

On a new class of fractional partial differential equations II

Abstract In this paper we continue to advance the theory regarding the Riesz fractional gradient in the calculus of variations and fractional partial differential equations begun in an earlier work of the same name. In particular, we here establish an L 1 {L^{1}} Hardy inequality, obtain further regularity results for solutions of certain fractional PDE, demonstrate the existence of minimizers for integral functionals of the fractional gradient with non-linear dependence in the field, and also establish the existence of solutions to corresponding Euler–Lagrange equations obtained as conditions of minimality. In addition, we pose a number of open problems, the answers to which would fill in some gaps in the theory as well as to establish connections with more classical areas of study, including interpolation and the theory of Dirichlet forms.

A Dirichlet form approach to MCMC optimal scaling

This paper shows how the theory of Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis–Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.

Non-symmetric distorted Brownian motion: Strong solutions, strong Feller property and&#x0D; 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.

Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

A Class of Lévy Driven SDEs and their Explicit Invariant Measures

We describe a class of explicit invariant measures for stochastic differential equations driven by Levy noise. We relate them to the corresponding Fokker Planck equation. In the symmetric case, we point out the relation with the theory of Dirichlet forms and generalized Schrodinger type operators.

Optimal transport, Cheeger energies and contractivity of dynamic transport distances in extended spaces

We introduce the setting of extended metric–topological measure spaces as a general “Wiener like” framework for optimal transport problems and nonsmooth metric analysis in infinite dimension.After a brief review of optimal transport tools for general Radon measures, we discuss the notions of the Cheeger energy, of the Radon measures concentrated on absolutely continuous curves, and of the induced “dynamic transport distances”. We study their main properties and their links with the theory of Dirichlet forms and the Bakry–Émery curvature condition, in particular concerning the contractivity properties and the EVI formulation of the induced Heat semigroup.

An Orthogonal-Polynomial Approach to First-Hitting Times of Birth\u2013Death Processes

In a recent paper in this journal, Gong, Mao and Zhang, using the theory of Dirichlet forms, extended Karlin and McGregor’s classical results on first-hitting times of a birth–death process on the nonnegative integers by establishing a representation for the Laplace transform {mathbb {E}}[e^{sT_{ij}}] of the first-hitting time T_{ij} for any pair of states i and j, as well as asymptotics for {mathbb {E}}[e^{sT_{ij}}] when either i or j tends to infinity. It will be shown here that these results may also be obtained by employing tools from the orthogonal-polynomial toolbox used by Karlin and McGregor, in particular associated polynomials and Markov’s theorem.

Dirichlet Form Theory and its Applications

Theory of Dirichlet forms is one of the main achievements in modern probability theory. It provides a powerful connection between probabilistic and analytic potential theory. It is also an effective machinery for studying various stochastic models, especially those with non-smooth data, on fractal-like spaces or spaces of infinite dimensions. The Dirichlet form theory has numerous interactions with other areas of mathematics and sciences. This workshop brought together top experts in Dirichlet form theory and related fields as well as promising young researchers, with the common theme of developing new foundational methods and their applications to specific areas of probability. It provided a unique opportunity for the interaction between the established scholars and young researchers.

New compactly supported spatiotemporal covariance functions from SPDEs

Potential theory and Dirichlet’s priciple constitute the basic elements of the well-known classical theory of Markov processes and Dirichlet forms. This paper presents new classes of fractional spatiotemporal covariance models, based on the theory of non-local Dirichlet forms, characterizing the fundamental solution, Green kernel, of Dirichlet boundary value problems for fractional pseudodifferential operators. The elements of the associated Gaussian random field family have compactly supported non-separable spatiotemporal covariance kernels admitting a parametric representation. Indeed, such covariance kernels are not self-similar but can display local self-similarity, interpolating regular and fractal local behavior in space and time. The associated local fractional exponents are estimated from the empirical log-wavelet variogram. Numerical examples are simulated for illustrating the properties of the space–time covariance model class introduced.

Probabilistic representations of solutions of elliptic boundary value problem and non-symmetric semigroups

In this paper, we use a probabilistic approach to show that there exists a unique, bounded continuous solution to the Dirichlet boundary value problem for a general class of second order non-symmetric elliptic operators L with singular coefficients, which does not necessarily have the maximum principle. The theory of Dirichlet forms and heat kernel estimates play a crucial role in our approach. A probabilistic representation of the non-symmetric semigroup {Tt}t≥0 generated by L is also given.

Equilibrium Diffusion on the Cone of Discrete Radon Measures

Let ${\mathbb {K}(\mathbb {R}^{d})}$ denote the cone of discrete Radon measures on $\mathbb {R}^{d}$ . There is a natural differentiation on $\mathbb {K}(\mathbb {R}^{d})$ : for a differentiable function $F:\mathbb {K}(\mathbb {R}^{d})\to \mathbb {R}$ , one defines its gradient $\nabla ^{\mathbb {K}}F$ as a vector field which assigns to each $\eta \in \mathbb {K}(\mathbb {R}^{d})$ an element of a tangent space $T_{\eta }(\mathbb {K}(\mathbb {R}^{d}))$ to $\mathbb {K}(\mathbb {R}^{d})$ at point η. Let $\phi :\mathbb {R}^{d}\times \mathbb {R}^{d}\to \mathbb {R}$ be a potential of pair interaction, and let μ be a corresponding Gibbs perturbation of (the distribution of) a completely random measure on $\mathbb {R}^{d}$ . In particular, μ is a probability measure on $\mathbb {K}(\mathbb {R}^{d})$ such that the set of atoms of a discrete measure $\eta \in \mathbb {K}(\mathbb {R}^{d})$ is μ-a.s. dense in $\mathbb {R}^{d}$ . We consider the corresponding Dirichlet form $$\mathcal{E}^{\mathbb{K}}(F,G)={\int}_{\mathbb K(\mathbb{R}^{d})}\langle\nabla^{\mathbb{K}} F(\eta), \nabla^{\mathbb{K}} G(\eta)\rangle_{T_{\eta}(\mathbb{K})}\,d\mu(\eta). $$ Integrating by parts with respect to the measure μ, we explicitly find the generator of this Dirichlet form. By using the theory of Dirichlet forms, we prove the main result of the paper: If d ≥ 2, there exists a conservative diffusion process on $\mathbb {K}(\mathbb {R}^{d})$ which is properly associated with the Dirichlet form $\mathcal {E}^{\mathbb {K}}$ .

Riemannian Ricci curvature lower bounds in metric measure spaces with 𝜎-finite measure

In a prior work of the first two authors with Savaré, a new Riemannian notion of a lower bound for Ricci curvature in the class of metric measure spaces ( X , d , m ) (X,\textsf {d},\mathbf {m}) was introduced, and the corresponding class of spaces was denoted by R C D ( K , ∞ ) RCD(K,\infty ) . This notion relates the C D ( K , N ) CD(K,N) theory of Sturm and Lott-Villani, in the case N = ∞ N=\infty , to the Bakry-Emery approach. In this prior work the R C D ( K , ∞ ) RCD(K,\infty ) property is defined in three equivalent ways and several properties of R C D ( K , ∞ ) RCD(K,\infty ) spaces, including the regularization properties of the heat flow, the connections with the theory of Dirichlet forms and the stability under tensor products, are provided. In the above-mentioned work only finite reference measures m \mathbf {m} have been considered. The goal of this paper is twofold: on one side we extend these results to general σ \sigma -finite spaces, and on the other we remove a technical assumption that appeared in the earlier work concerning a strengthening of the C D ( K , ∞ ) CD(K,\infty ) condition. This more general class of spaces includes Euclidean spaces endowed with Lebesgue measure, complete noncompact Riemannian manifolds with bounded geometry and the pointed metric measure limits of manifolds with lower Ricci curvature bounds.

Matrix-valued Bessel processes

This paper introduces a matrix analog of the Bessel processes, taking values in the closed set $E$ of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map $x\mapsto x^\top x$. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function $w(x)=|\det x|^\alpha$ is a weight function in the Muckenhoupt $A_p$ class for $-1<\alpha\le 0$ ($p=1$) and $-1<\alpha<p-1$ ($p>1$). The set of matrices of co-rank at least two has zero capacity with respect to the measure $m(dx)=|\det x|^\alpha dx$ if $\alpha>-1$, and if $\alpha\ge 1$ this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) $E$ with Neumann boundary conditions. The highly non-convex, non-Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods.

Mixed boundary value problems of semilinear elliptic PDEs and BSDEs with singular coefficients

In this paper, we prove that there exists a unique weak (Sobolev) solution to the mixed boundary value problem for a general class of semilinear second order elliptic partial differential equations with singular coefficients. Our approach is probabilistic. The theory of Dirichlet forms and backward stochastic differential equations with singular coefficients and infinite horizon plays a crucial role.

On countably skewed Brownian motion with accumulation point

In this work we connect the theory of symmetric Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.