Notes of Anton Thalmaier’s lecture nr 3

1. Stochastic flows of diffeomorphisms

We continue our study of SDE ${dX=A_0(X)dt+\sum A_i(X)\cdot dB^i}$. Up to now, the starting point ${x}$ was fixed. Now we exploit the dependance on ${x}$.

1.1. Random continuous paths of diffeomorphisms

Let us introduce the random set ${M_t(\omega)=\{x\in M\,;\,\zeta(x)(\omega)>t\}}$ of starting points whose trajectory is still alive at time ${t}$. Then

• ${M_t(\omega)}$ is open (in fact, the lifetime ${\zeta(\cdot)(\omega)}$ is lower semi-continuous in ${x}$).
• ${X_t(\cdot)(\omega):M_t(\omega)\rightarrow R_t(\omega)}$ is a diffeomorphism onto an open subset of ${M}$.
• ${s\mapsto X_s(\cdot)(\omega)}$ is continuous: ${[0,t]\rightarrow C^{\infty}(M_t(\omega),M)}$.

Furthermore, under mild growth conditions on vectorfields and their derivatives (for instance, if ${M}$ is compact), ${X_t(\cdot)(\omega)\in Diffeo(M)}$ for all ${t}$.

Consider the tangent flow ${U={X_t}_*}$ on ${TM}$. It solves the formally differentiated SDE

$\displaystyle \begin{array}{rcl} dU=\sum (DA_i)_X U\cdot dZ^i. \end{array}$

1.2. Crucial observation

Let us transport a vectorfield ${V}$ under our stochastic flow. We get a random vectorfield ${{X_t}^{-1}_{*}V}$. This means that, for a test function ${f}$,

$\displaystyle \begin{array}{rcl} ({X_t}^{-1}_{*}V)(f)=(V(f\circ X_t^{-1}))\circ X_t. \end{array}$

Maillavin’s covariance matrix is defined as follows. For ${t>0}$,

$\displaystyle \begin{array}{rcl} C_t(x)=\sum_{i=1}^r\int_{0}^{t}({X_s}^{-1}_{*}A_i)\otimes({X_s}^{-1}_{*}A_i)_X\, dt. \end{array}$

This is a random smooth section of ${TM\otimes TM}$ over ${M_t}$. We shall see later that the condition we need to make this nondegenerate is Hörmander’s condition.

On may view

$\displaystyle \begin{array}{rcl} ({X_s}^{-1}_{*}A)_X:{\mathbb R}^r \rightarrow T_X M \end{array}$

as a linear map from ${{\mathbb R}^r}$ to ${T_X M}$. Its adjoint is a linear map from ${T_X^* M}$ to ${{\mathbb R}^r}$. Then ${C_t(x)}$ may be viewed as an endomorphism of ${T_x M}$.

Lemma 1 The SDE satisfied by ${{X_t}^{-1}_{*}V}$ is

$\displaystyle \begin{array}{rcl} d({X_t}^{-1}_{*}V)=\sum_{i=0}^{r}({X_t}^{-1}_{*}[A_i,V])_X \cdot dZ^i. \end{array}$

In particular, if ${V}$ commutes with vectorfields ${A_i}$, ${{X_t}^{-1}_{*}V=V}$.

2. Stochastic flows and hypoellipticity

We assume that all constant coefficient combinations of the ${A_i}$ are complete. The flow defines two canonical measures,

• The distribution of ${X_t(x)}$, ${P_t(x,dy)=P\{X_t(x)\in dy\}}$,
• Green’s measure ${G_\lambda(x,dy)=\int_{0}^{\infty}P_t(x,dy)\,dt}$.

Let us study the following Dirichlet boundary problem

$\displaystyle \begin{array}{rcl} -Lu+ku&=&f \textrm{ on }D,\\ u_{\partial D}&=&\phi. \end{array}$

The solution takes the following form (Feyman-Kac formula).

$\displaystyle \begin{array}{rcl} u(x)=\mathop{\mathbb E}(\phi(X_{\tau_D})\exp(-\int_{0}^{\tau_D}k(X_s)\,ds)+\int_{0}^{\tau_D}f(X_s)\exp(\int_{0}^{s}k(X_r)\,dr)\,ds). \end{array}$

2.1. Hörmander’s condition

Question: When do ${P_t(x,dy)}$ and ${G_\lambda(x,dy)}$ have a density ?

Let

• ${\mathcal{L}}$ denote the Lie algebra generated by the vectorfield ${A_i}$,
• ${\mathcal{B}}$ the Lie algebra generated by ${A_1,\ldots,A_r}$ only,
• ${\mathcal{J}}$ by ${A_1,\ldots,A_r}$ and brackets ${[A_0,A_i]}$,
• ${\hat{\mathcal{L}}}$ by ${A_0+\partial_t}$ and ${A_1,\ldots,A_r}$ on ${M\times{\mathbb R}}$.

Hörmander’s theorem states

• hypoellipticity of ${L}$ under ${\mathcal{L}(x)=T_xM}$,
• hypoellipticity of ${L+\partial_t}$ under ${\hat{\mathcal{L}}(x)=T_{x,t}M\times{\mathbb R}}$.

It follows that

• Under ${\mathcal{L}(x)=T_xM}$, ${G_\lambda(x,dy)=g_\lambda(x,y)\,dy}$,
• Under ${\hat{\mathcal{L}}(x)=T_{x,t}M\times{\mathbb R}}$, ${P_t(x,dy)=p_t(x,y)\,dy}$,

where the densities are smooth.

2.2. A probabilistic proof of hypoellipticity ?

In 1970, in his Kyoto lectures, Paul Maillavin proposed a toolbox to prove this, called Maillavin Calculus. This calculus deals with infinite dimensional path spaces.

Instead, I will describe a more direct root. The existence of smooth densities ${g_\lambda}$ and ${p_t}$ in turn imply hypoellipticity, so it suffices to prove this.

Theorem 2 Assume that ${A_i}$ and there derivatives satisfy suitable growth conditions. Assume that the bilinear form ${C_t(x)}$ is non-degenerate and

$\displaystyle \begin{array}{rcl} |C_t(x)|^{-1}\in L^p \end{array}$

for all ${p\geq 1}$. Then ${P_t(x,dy)=p_t(x,y)\,dy}$ with a smooth density ${p_t(x,y)}$.