On a martingale method for symmetric diffusion processes and its applications

a martingale method for symmetric diffusion processes andits application

Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

Let E be an unbounded open (or closed) domain in Euclidean space of dimension greater or equal to two. We present conservativeness criteria for (possibly reflected) diffusions with state space E that are associated to fairly general perturbed divergence form operators. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons-Zheng decomposition in the symmetric case.Comment: Corrected typos, minor modification

Explosion by Killing and Maximum Principle in Symmetric Markov Processes

Keller and Lenz \cite{KL} define a concept of {\it stochastic completeness at infinity} (SCI) for a regular symmetric Dirichlet form (\cE,\cF). We show that (SCI) can be characterized probabilistically by using the predictable part $\zeta^p$ of the life time $\zeta$ of the symmetric Markov process $X=({\bf P}_x,X_t)$ generated by (\cE,\cF), that is, (SCI) is equivalent to \bfP_x(\zeta=\zeta^p<\infty)=0. We define a concept, {\it explosion by killing} (EK), by \bfP_x(\zeta=\zeta^i<\infty)=1. Here $\zeta^i$ is the totally inaccessible part of $\zeta$. We see that (EK) is equivalent to (SCI) and \bfP_x(\zeta=\infty)=1. Let $X^{\rm res}$ be the {\it resurrected process} generated by the {\it resurrected form}, a regular Dirichlet form constructed by removing the killing part from (\cE, \cF). Extending work of Masamune and Schmidt (\cite{MS}), we show that (EK) is also equivalent to the ordinary conservation property of time changed process of $X^{\rm res}$ by $A^k_t$, where the $A^k_t$ is the positive continuous additive functional in the Revuz correspondence to the killing measure $k$ in the Beurling-Deny formula (Theorem \ref{ma-sh}). We consider the maximum principle for Schr\"odinger-type operator \cL^\mu=\cL-\mu. Here \cL is the self-adjoint operator associated with (\cE,\cF) %with non-local part and $\mu$ is a Green-tight Kato measure. Let $\lambda(\mu)$ be the principal eigenvalue of the trace of (\cE,\cF) relative to $\mu$. We prove that if (EK) holds, then $\lambda(\mu)>1$ implies a Liouville property that every bounded solution to \cL^\mu u=0 is zero quasi-everywhere and that the {\it refined maximum principle} in the sense of Berestycki-Nirenberg-Varadhan \cite{BNV} holds for \cL^\mu if and only if $\lambda(\mu)>1$ (Theorem \ref{RMP})

OPTIMAL HARDY-TYPE INEQUALITIES FOR SCHRÖDINGER FORMS

We give a method to construct a critical Schrödinger form from the subcritical Schrödinger form by subtracting a suitable positive potential. The method enables us to obtain optimal Hardy-type inequalities