Verification Theorems for Stochastic Optimal Control Problems via a Time Dependent Fukushima - Dirichlet Decomposition

On $L^p$ Liouville theorems for Dirichlet forms

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.

Weak Dirichlet processes with a stochastic control perspective

The motivation of this paper is to prove verification theorems for stochastic optimal control of finite dimensional diffusion processes without control in the diffusion term, in the case that the value function is assumed to be continuous in time and once differentiable in the space variable ($C^{0,1}$) instead of once differentiable in time and twice in space ($C^{1,2}$), like in the classical results. For this purpose, the replacement tool of the Itô formula will be the Fukushima-Dirichlet decomposition for weak Dirichlet processes. Given a fixed filtration, a weak Dirichlet process is the s…

Class of smooth functions in Dirichlet spaces

Given a regular Dirichlet form $(\mathcal{E},\mathcal{F})$ on a fixed domain $E$ of $\mathbb{R}^d$, we first indicate that the basic assumption $C_c^\infty(E)\subset \mathcal{F}$ is equivalent to the fact that each coordinate function $f^i(x)=x_i$ locally belongs to $\mathcal{F}$ for $1\leq i\leq d$. Our research starts from these two different viewpoints. On one hand, we shall explore when $C_c^\infty(E)$ is a special standard core of $\mathcal{F}$ and give some useful characterizations. On the other hand, we shall describe the Fukushima's decompositions of $(\mathcal{E},\mathcal{F})$ with re…

道格拉斯加权狄利克雷空间：分析与概率方面

Douglas Weighted Dirichlet Spaces: Analytic and Probabilistic Aspects

本文引入道格拉斯加权狄利克雷空间，证明其再生核是Szego核在1预解式下的像，发展相关位势理论，得到关联对称纯跳Hunt过程，经典情形下为缠绕柯西过程。

Stochastic calculus over symmetric Markov processes without time reversal

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain cl…

Infinite dimensional weak Dirichlet processes, stochastic PDEs and optimal control

The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of "weak Dirichlet process" in this context. Such a process $\X$, taking values in a Hilbert space $H$, is the sum of a local martingale and a suitable "orthogonal" process. The new concept is shown to be useful in several contexts and directions. On one side, the mentioned decomposition appears to be a substitute of an Itô type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and…

具有勒贝格测度零集合上可渗透粘性行为的扭曲布朗运动的构造

Construction of distorted Brownian motion with permeable sticky behaviour on sets with Lebesgue measure zero

本文构造了在勒贝格测度零集合上具有可渗透粘性行为的扭曲布朗运动，通过梯度狄利克雷形式定义过程，并证明其生成元表示、不可约性、常返性及正逗留时间。

Analysis of Stochastic Quantization for the fractional Edwards Measure

We analyse a diffusion process whose invariant measure is the fractional polymer or Edwards measure for fractional Brownian motion in dimension $d\in\mathbb{N}$ with Hurst parameter $H\in(0,1)$ fulfilling $dH < 1$. We make use of a construction of the diffusion via Dirichlet form techniques in infinite dimensional (Gaussian) analysis. By providing a Fukushima decomposition for the stochastic quantization of the fractional Edwards measure we prove that the constructed process solves weakly a stochastic differential equation in infinite dimension for quasi-all starting points. Moreover, the solu…

Verification by stochastic Perron's method in stochastic exit time control problems

We apply the Stochastic Perron method, created by Bayraktar and Sîrbu, to a stochastic exit time control problem. Our main assumption is the validity of the Strong Comparison Result for the related Hamilton-Jacobi-Bellman (HJB) equation. Without relying on Bellman's optimality principle we prove that inside the domain the value function is continuous and coincides with a viscosity solution of the Dirichlet boundary value problem for the HJB equation.