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 1 Trajectories 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 2 If , 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 1 Given -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 .