Malliavin Calculus for Degenerate Diffusions

Let $(W,H,\mu)$ 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 $\sigma: \R^n\to \R^n\otimes \R^d$, $b: \R^n\to \R^n$, $B$ is an $\R^d$-valued Brownian motion. We suppose that $b$ and $\sigma$ are Lipschitz. Let $P(x)$ be the orthogonal projection from $\R^d$ to its closed subspace $\sigma(x)^\star(\R^n)$, 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 $L^p(\nu)$, where $\nu$ represents the law of the above diffusion process, i.e., $\nu=X(x)(\mu)$, 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 $L^2(\nu)$-functionals, the logarithmic Sobolev inequality for $\nu$. 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.