**1. -diffusions and the heat equation **

The problem: given a continuous function , find a solution , , of

Assume is an -diffusion with lifetime . Then is an -diffusion, where .

Hypothesis: assume that the lifetime is almost surely infinite.

Assume a bounded solution exists. We show uniqueness and provide a formula for . For , consider the restriction of to the time interval . Reverse time. The following process

is a local martingale. I do as if it were a martingale, to save time. Since expectations do not depend on time,

This proves uniqueness of bounded solutions to the heat equation.

** 1.1. Regularity of trajectories **

Remark 1Trajectories cannot be regular, even rectifiable.

Indeed, the It\^o bracket of two processes vanishes provided one of them has locally rectifiable trajectories.

consider the *carré du champ* operator

For instance, if ,

Hence vanishes if and only if is first order. Here is the formula for the It\^o bracket,

it does not vanish in general.

Nevertheless, expectation kills the fluctuations.

**2. Construction of stochastic flows **

** 2.1. Stochastic differential equation **

Let be a finite dimensional vectorspace. A stochastic differential equation (SDE) on a manifold is the data of

- a continuous semi-martingale taking values in (i.e. where has bounded variation and is a martingale),
- a vectorbundle morphism (in other words, a vectorfield depending linearly on a vector parameter ).

The equation to be solved is formally written

It intuitively means that driving process and matrix tell us in which direction to move. More formally, a solution of the SDE with initial condition is an adapted continuous process such that, for smooth, compactly supported ,

The dot denotes a Stratonovich integral. The integral can be rewritten

Remark 2If , are real semi-martingales, then the formula relating the Stratonovich and It\^o integrals is

Note that .

Stratonovich’s notation makes It\^o’s formula shorter,

Theorem 1Given -measurable random variable , an SDE has a unique maximal solution.

Uniqueness means that an other solution with the same initial condition coincides a.s. on the common life time interval.

** 2.2. Main example **

In , independant Brownian motions on . Then the equation becomes

i.e. for every test function,

The middle term is a martingale, so we get, for ,

**3. Control theory and support theorems **

** 3.1. Support theorem **

We shall relate the SDE

with independant Brownian driving processes, with the control problem

Let denote the stochastic flow and the endpoint map at time . Let denote the forward orbit of , i.e. the set of all , all , all possible controls. Write

where , the orbit at time , is the image of the endpoint map at time . View the process as a probability measure on the space of continuous paths. Consider its distribution at time , . Let denote the Green measure with parameter ,

Theorem 2 (Stroock-Varadhan)1. Path space: the support of is the closure of the forward orbit .2. State space: the support of is the closure of the time orbit . The support of Green’s measure is the closure of the forward orbit .