论文标题
变性扩散的Malliavin演算
Malliavin Calculus for Degenerate Diffusions
论文作者
论文摘要
令$（w，h，μ）$为$ \ r^d $上的经典维纳空间。 Assume that $X=(X_t(x))$ is a diffusion process satisfying the stochastic differential equation with diffusion and drift coefficients $σ: \R^n\to \R^n\otimes \R^d$, $b: \R^n\to \R^n$, $B$ is an $\R^d$-valued Brownian motion.我们认为$ b $和$σ$是Lipschitz。 Let $P(x)$ be the orthogonal projection from $\R^d$ to its closed subspace $σ(x)^\star(\R^n)$, assuming that $x\to P(x)$ is continuously differentiable, we construct a covariant derivative $\hat{\nabla}$ on the paths of the diffusion process, along the elements of the Cameron-Martin space and证明该导数可在$ l^p（ν）$上关闭，其中$ν$代表上述扩散过程的定律，即$ν= x（x）（x）（μ）$，函数$ w \ to x_ \ cdot（w，x）$下的wiener度量的图像。我们研究了该操作员的伴随，并证明了几个结果：$ l^2（ν）$ - 功能的表示定理，$ν$的对数sobolev不平等。随着这些结果的应用，通过使用协方差衍生物给出了对数Sobolev不等式在戴森布朗运动路径空间上的证明。然后，我们解释了如何使用该理论来得出由$ t = 1 $的扩散过程的半群和固定起点定义的措施的功能不平等。最后，我们表明，由于有条件的独立性结果，这也是有条件措施的这些不平等，这是扩散过程退化的结果。
Let $(W,H,μ)$ be the classical Wiener space on $\R^d$. Assume that $X=(X_t(x))$ is a diffusion process satisfying the stochastic differential equation with diffusion and drift coefficients $σ: \R^n\to \R^n\otimes \R^d$, $b: \R^n\to \R^n$, $B$ is an $\R^d$-valued Brownian motion. We suppose that $b$ and $σ$ are Lipschitz. Let $P(x)$ be the orthogonal projection from $\R^d$ to its closed subspace $σ(x)^\star(\R^n)$, assuming that $x\to P(x)$ is continuously differentiable, we construct a covariant derivative $\hat{\nabla}$ on the paths of the diffusion process, along the elements of the Cameron-Martin space and prove that this derivative is closable on $L^p(ν)$, where $ν$ represents the law of the above diffusion process, i.e., $ν=X(x)(μ)$, the image of the Wiener measure under the function $w\to X_\cdot(w,x)$. We study the adjoint of this operator and we prove several results: representation theorem for $L^2(ν)$-functionals, the logarithmic Sobolev inequality for $ν$. As applications of these results the proof of the Logarithmic Sobolev inequality on the path space of Dyson's Brownian motion is given by using the covariant derivative. We then explain how to use this theory for deriving the functional inequalities for the measures defined by the semigroups of the diffusion process at the time $t=1$ and with fixed starting point. Finally we show that one can obtain also these inequalities for the conditional measures due to a conditional independence result which is a consequence of the degeneracy of the diffusion process.
