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)}.

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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