Quasi-regular Dirichlet forms and the obstacle problem for elliptic equations with measure data

Feynman-Kac Formulas for Solutions to Degenerate Elliptic and Parabolic Boundary-Value and Obstacle Problems with Dirichlet Boundary Conditions

We prove Feynman-Kac formulas for solutions to elliptic and parabolic boundary value and obstacle problems associated with a general Markov diffusion process. Our diffusion model covers several popular stochastic volatility models, such as the Heston model, the CEV model and the SABR model, which are widely used as asset pricing models in mathematical finance. The generator of this Markov process with killing is a second-order, degenerate, elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the $2α$-power of the distance to the boundary of the…

一种树状分形Dirichlet空间，位于强椭圆Harnack不等式与弱椭圆Harnack不等式之间

A tree-like fractal Dirichlet space lying between strong and weak elliptic Harnack inequalities

本文构建了一个自相似分形树状结构，并赋予其正则自相似Dirichlet形式，展示了均退出时间相对于典型度量球的异常行为，且在适当选择的自相似测度下，弱椭圆Harnack不等式成立而强不等式不成立。

Dirichlet forms and semilinear elliptic equations with measure data

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are giv…

A classical Perron method for existence of smooth solutions to boundary value and obstacle problems for degenerate-elliptic operators via holomorphic maps

We prove existence of solutions to boundary value problems and obstacle problems for degenerate-elliptic, linear, second-order partial differential operators with partial Dirichlet boundary conditions using a new version of the Perron method. The elliptic operators considered have a degeneracy along a portion of the domain boundary which is similar to the degeneracy of a model linear operator identified by Daskalopoulos and Hamilton (1998) in their study of the porous medium equation or the degeneracy of the Heston operator (1993) in mathematical finance. Existence of a solution to the Dirichl…

Asymptotics in the Dirichlet Problem for Second Order Elliptic Equations with Degeneration on the Boundary

We study small perturbations of the Dirichlet problems for second order elliptic equations that degenerate on the boundary. The limit of the solution, as the perturbation tends to zero, is calculated. The result is based on a certain asymptotic self-similarity near the boundary, which holds in the generic case. In the last section, we briefly consider the stabilization of solutions to the corresponding parabolic equations with a small parameter. Metastability effects arise in this case: the asymptotics of the solution depends on the time scale. Initial-boundary value problem with the Neumann b…

Construction of $\mathcal L^p$-strong Feller Processes via Dirichlet Forms and Applications to Elliptic Diffusions

We provide a general construction scheme for $\mathcal L^p$-strong Feller processes on locally compact separable metric spaces. Starting from a regular Dirichlet form and specified regularity assumptions, we construct an associated semigroup and resolvents of kernels having the $\mathcal L^p$-strong Feller property. They allow us to construct a process which solves the corresponding martingale problem for all starting points from a known set, namely the set where the regularity assumptions hold. We apply this result to construct elliptic diffusions having locally Lipschitz matrix coefficients …

Dirichlet's and Thomson's principles for non-selfadjoint elliptic operators with application to non-reversible metastable diffusion processes

We present two variational formulae for the capacity in the context of non-selfadjoint elliptic operators. The minimizers of these variational problems are expressed as solutions of boundary-value elliptic equations. We use these principles to provide a sharp estimate for the transition times between two different wells for non-reversible diffusion processes. This estimate permits to describe the metastable behavior of the system.

C^{1,1} regularity for degenerate elliptic obstacle problems

The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator, where the degeneracy in the operator symbol is proportional to the distance to the boundary of the half-plane. In mathematical finance, solutions to the obstacle problem for the elliptic Heston operator correspond to value functions for p…

Existence, uniqueness, and global regularity for degenerate elliptic obstacle problems in mathematical finance

The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order degenerate elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the boun…

通过Dirichlet形式和预解式研究定常Fokker-Planck方程解的局部椭圆正则性

Local elliptic regularity for solutions to stationary Fokker-Planck equations via Dirichlet forms and resolvents

本文针对具一般系数的定常Fokker-Planck方程解（L²密度测度），利用Dirichlet形式和预解式，结合椭圆正则性结果，证明其密度具H¹·²正则性与Hölder连续性，突显相关理论在解正则性分析中的核心作用。

通过Dirichlet形式扰动实现紧群上的次椭圆扩散

Sub-elliptic diffusions on compact groups via Dirichlet form perturbation

将有限维次椭圆理论推广到无限维紧群，通过距离比较不等式建立Dirichlet形式控制，进而传递热核性质，并以SU(2)无限积为例验证。