## Notes of Anton Thalmayer’s lecture nr 2

1. ${L}$-diffusions and the heat equation

The problem: given a continuous function ${f}$, find a solution ${u=u(t,x)}$, ${t\geq 0}$, of

$\displaystyle \begin{array}{rcl} \frac{\partial u}{\partial t}&=&Lu,\\ u(0,\cdot)&=&f. \end{array}$

Assume ${X_t(x)}$ is an ${L}$-diffusion with lifetime ${\zeta(x)}$. Then ${(t,X_t(x))}$ is an ${L'}$-diffusion, where ${L'=\partial_t +L}$.

Hypothesis: assume that the lifetime is almost surely infinite.

Assume a bounded solution ${u(t,x)}$ exists. We show uniqueness and provide a formula for ${u}$. For ${t>0}$, consider the restriction of ${u}$ to the time interval ${[0,t]}$. Reverse time. The following process

$\displaystyle \begin{array}{rcl} u(t-s,X_s(x))-u(t,x)-\int_{0}^{t}(\partial_t + L)u(t-r,X_r(x))\,dr \end{array}$

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

$\displaystyle \begin{array}{rcl} u(t,x)=\mathop{\mathbb E}(u(t-s,X_s(x)))=\mathop{\mathbb E}(u(0,X_t(x)))=\mathop{\mathbb E}(f(X_t(x))). \end{array}$

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

$\displaystyle \begin{array}{rcl} \Gamma(f,g)=\frac{1}{2}(L(fg)-fL(g)-gL(f)). \end{array}$

For instance, if ${L=A_0+\sum A_i^2}$,

$\displaystyle \begin{array}{rcl} \Gamma(f,g)=\sum A_i(f)A_i(g). \end{array}$

Hence ${\Gamma}$ vanishes if and only if ${L}$ is first order. Here is the formula for the It\^o bracket,

$\displaystyle \begin{array}{rcl} [f(X),g(X)]=2\int_{0}^{t}\Gamma(f,g)(X_r)\,dr, \end{array}$

it does not vanish in general.

Nevertheless, expectation kills the fluctuations.

2. Construction of stochastic flows

2.1. Stochastic differential equation

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

• a continuous semi-martingale ${Z_t}$ taking values in ${E}$ (i.e. ${Z_t=N_t+A_t}$ where ${A_t}$ has bounded variation and ${N_t}$ is a martingale),
• a vectorbundle morphism ${A:M\times E\rightarrow TM}$ (in other words, a vectorfield depending linearly on a vector parameter ${e\in E}$).

The equation to be solved is formally written

$\displaystyle \begin{array}{rcl} dX=A(X)\circ dZ. \end{array}$

It intuitively means that driving process ${Z}$ and matrix ${A}$ tell us in which direction to move. More formally, a solution of the SDE with initial condition ${X_0}$ is an adapted continuous process such that, for smooth, compactly supported ${f}$,

$\displaystyle \begin{array}{rcl} f(X_t)=f(X_0)+\int_{0}^{t}(df)_{X_s}A(X_s)\cdot dZ_s. \end{array}$

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

$\displaystyle \begin{array}{rcl} \int_{0}^{t}\sum A_i (f)(X_s)\cdot dZ_s. \end{array}$

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

$\displaystyle \begin{array}{rcl} \int_{0}^{t}Y\cdot dZ=\int_{0}^{t}YdZ+\frac{1}{2}[Y,Z]_t . \end{array}$

Note that ${d(Y\cdot Z)=Y\cdot dZ+Z\cdot dY}$.

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

$\displaystyle \begin{array}{rcl} f(X_t)&=&f(X_0)+\sum\int_{0}^{t}D_i f(X_s)\,dX^i_s+\frac{1}{2}\sum\int_{0}^{t}D_i D_j f(X_s)d[Xî_s,Xj_s]_s\\ &=&f(X_0)+\sum\int_{0}^{t}\langle\nabla f(X_s),\cdot dX_s\rangle \end{array}$

Theorem 1 Given ${\mathcal{F}_0}$-measurable random variable ${X_0}$, 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 ${{\mathbb R}^n}$, ${Z_t=(t,B^1_t,\ldots,B^r_t)}$ independant Brownian motions on ${{\mathbb R}}$. Then the equation becomes

$\displaystyle \begin{array}{rcl} dX=A_0(X)dt+\sum A_i(X)\cdot dB^i, \end{array}$

i.e. for every test function,

$\displaystyle \begin{array}{rcl} d(f\cdot X)&=&A_0(f)(X)+\sum A_if(X)\cdot dB^i\\ &=&A_0(f)(X)+\sum A_if(X)dB^i+\frac{1}{2}\sum d[A_if(X),B^i]\\ &=&A_0(f)(X)+\sum A_if(X)dB^i+\frac{1}{2}(\sum A_i^2)(f)(X). \end{array}$

The middle term is a martingale, so we get, for ${L=A_0+ \frac{1}{2}\sum A_i^2}$,

$\displaystyle \begin{array}{rcl} f(X_t)-f(X_0)-\int_{0}^{t}(Lf)(X_s)\,ds\quad\textrm{ is a martingale.} \end{array}$

3. Control theory and support theorems

3.1. Support theorem

We shall relate the SDE

$\displaystyle \begin{array}{rcl} dX=A_0(X)dt+\sum A_i(X)\cdot dB^i, \end{array}$

with independant Brownian driving processes, with the control problem

$\displaystyle \begin{array}{rcl} \dot{x}(t)=A_0(x(t))dt+\sum A_i(x(t))u^i(t). \end{array}$

Let ${X_t(x)}$ denote the stochastic flow and ${\phi_t(x,u)}$ the endpoint map at time ${t}$. Let ${O^+(x)}$ denote the forward orbit of ${x}$, i.e. the set of all ${\phi_t(x,u)}$, all ${t\geq 0}$, all possible controls. Write

$\displaystyle \begin{array}{rcl} O^+(x)=\bigcup_t O^t(x). \end{array}$

where ${O^t(x)}$, the orbit at time ${t}$, is the image of the endpoint map at time ${t}$. View the process as a probability measure ${P_x}$ on the space of continuous paths. Consider its distribution at time ${t}$, ${P_{t,x}}$. Let ${G_\lambda}$ denote the Green measure with parameter ${\lambda}$,

$\displaystyle \begin{array}{rcl} G_\lambda(x,\cdot)=\int_{0}^{\infty}e^{-\lambda t}P_{x,t}(\cdot)\,dt. \end{array}$

Theorem 2 (Stroock-Varadhan) 1. Path space: the support of ${P_x}$ is the closure of the forward orbit ${O^+(x)}$.

2. State space: the support of ${P_{x,t}}$ is the closure of the time ${t}$ orbit ${O^+(x)}$. The support of Green’s measure ${G_\lambda}$ is the closure of the forward orbit ${O^+(x)}$.