Regular subspaces of skew product diffusions

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space ${E_{1}\times E_{2}}$ and expressed as$X_{t}=(X^{1}_{t},X^{2}_{A_{t}}),\quad t\geq 0,$where ${X^{i}}$ is a symmetric diffusion on ${E_{i}}$ for ${i=1,2}$, and A is a positive continuous additive functional of ${X^{1}}$. One of our main results indicates that any skew product type regular subspace of X, say$Y_{t}=(Y^{1}_{t},{Y^{2}_{\tilde{A}_{t}}}),\quad t\geq 0,$can be characterized as follows: the associated smooth measure of ${\tilde{A}}$ is equal to that of A, and ${Y^{i}}$ corresponds to a regular subspace of ${X^{i}}$ for ${i=1,2}$. Furthermore, we shall make some discussions on rotationally invariant diffusions on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$, which are special skew product diffusions on ${(0,\infty)\times S^{d-1}}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ to a new regular Dirichlet form on ${\mathbb{R}^{d}}$. More precisely, fix a regular Dirichlet form ${(\mathcal{E,F}\kern 0.569055pt)}$ on the state space ${\mathbb{R}^{d}}$. Its part Dirichlet form on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is denoted by ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. Let ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$ be a regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$. We want to find a regular subspace ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ of ${(\mathcal{E,F}\kern 0.569055pt)}$ such that the part Dirichlet form of ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ on ${\mathbb{R}^{d}\setminus\{{\mathbf{0}}\}}$ is exactly ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. If ${(\tilde{\mathcal{E}},\tilde{\mathcal{F}}\kern 0.569055pt)}$ exists, we call it a regular extension of ${(\tilde{\mathcal{E}}^{0},\tilde{\mathcal{F}}{}^{0})}$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of ${(\mathcal{E}^{0},\mathcal{F}{}^{0})}$ has a unique regular extension.

Teasing Apart Two Trees

A widely studied model for generating binary sequences is to ‘evolve’ them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially ‘easier’ (in terms of the sequence length needed) than determining the true tree. The key tool is a new and near-tight Ramsey-type result for binary trees.

Diffusion processes with non-smooth diffusion coefficients and their density functions

SynopsisDenote by Xt an n-dimensional symmetric Markov process associated with an elliptic operatorwhere (aij) is a bounded measurable uniformly positive definite matrix-valued function of x. Let f(x, t) be a measurable function defined on Rn × [0, 1]. In this paper, we prove that f(Xt, t) is a regular Dirichlet process if and only if the following two conditions are satisfied:(i) For almost every and (ii) Let be a sequence of subdivisions of [0,1] so thatThenAs an application of the above result, we prove the following fact: Let p(y, t) be the probability density of the diffusion process Yt, associated with the elliptic operatorwhere (bi) are bounded measurable functions of x and we suppose that . Then, p(Yt, t) is a regular Dirichlet process and therefore p(.,.) satisfies (i) and (ii).