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)}.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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