Partial differential operator in Hörmander form

Let ${\displaystyle A\in \Gamma (TM)}$ be a vector field, understood as a derivation by the ${\displaystyle C^{\infty }(M)}$-isomorphism

${\displaystyle \Gamma (TM)\to \operatorname {Der} _{\mathbb {R} }C^{\infty }(M),\quad A\mapsto (f\mapsto Af)}$

for some ${\displaystyle f\in C^{\infty }(M)}$. The map ${\displaystyle Af:M\to \mathbb {R} }$ is defined by ${\displaystyle Af(x):=A_{x}(f)}$. For the composition, we set ${\displaystyle A^{2}:=A(A(f))}$ for some ${\displaystyle f\in C^{\infty }(M)}$.

A partial differential operator (PDO) ${\displaystyle L:C^{\infty }(M)\to C^{\infty }(M)}$ is given in Hörmander form if and only there are vector fields ${\displaystyle A_{0},A_{1},\dots ,A_{r}\in \Gamma (TM)}$ and ${\displaystyle L}$ can be written in the form

${\displaystyle L=A_{0}+\sum \limits _{i=1}^{r}A_{i}^{2}}$.

Flow process

Let ${\displaystyle L}$ be a PDO in Hörmander form on ${\displaystyle M}$ and ${\displaystyle x\in M}$ a starting point. An adapted and continuous ${\displaystyle M}$-valued process ${\displaystyle X}$ with ${\displaystyle X_{0}=x}$ is called a flow process to ${\displaystyle L}$ starting in ${\displaystyle x}$, if for every test function ${\displaystyle f\in C_{c}^{\infty }(M)}$ and ${\displaystyle t\in \mathbb {R} _{+}}$ the process

${\displaystyle N(f)_{t}:=f(X_{t})-f(X_{0})-\int _{0}^{t}Lf(X_{r})\mathrm {d} r}$

is a martingale, i.e.

${\displaystyle \mathbb {E} \left(N(f)_{t}\mid {\mathcal {F}}_{s}\right)=N(f)_{s},\quad \forall s\leq t}$.

Remark

For a test function ${\displaystyle f\in C_{c}^{\infty }(M)}$, a PDO ${\displaystyle L}$ in Hörmander form and a flow process ${\displaystyle X_{t}^{x}}$ (starting in ${\displaystyle x}$) also holds the flow equation, but in comparison to the deterministic case only in mean

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}\mathbb {E} f(X_{t}^{x})=\mathbb {E} \left[Lf(X_{t}^{x})\right]}$.

and we can recover the PDO by taking the time derivative at time 0, i.e.

${\displaystyle \left.{\frac {\mathrm {d} }{\mathrm {d} t}}\right|_{t=0}\mathbb {E} f(X_{t}^{x})=Lf(x)}$.

Lifetime and explosion time

Let ${\displaystyle \emptyset \neq U\subset \mathbb {R} ^{n}}$ be open und ${\displaystyle \xi >0}$ a predictable stopping time. We call ${\displaystyle \xi }$ the lifetime of a continuous semimartingale ${\displaystyle X=(X_{t})_{0\leq t<\xi }}$ on ${\displaystyle U}$ if

• there is a sequence of stopping times ${\displaystyle (\xi _{n})}$ with ${\displaystyle \xi _{n}\nearrow \xi }$, such that ${\displaystyle \xi _{n}<\xi }$ ${\displaystyle \mathbb {P} }$-almost surely on ${\displaystyle \{0<\xi <\infty \}}$.
• the stopped process ${\displaystyle (X_{t\wedge \xi _{n}})}$ is a semimartingale.

Moreover, if ${\displaystyle X_{\xi _{n}(\omega )}\to \partial U}$ for almost all ${\displaystyle \omega \in \{\xi <\infty \}}$, we call ${\displaystyle \xi }$ explosion time.

A flow process ${\displaystyle X}$ can have a finite lifetime ${\displaystyle \xi }$. By this we mean that ${\displaystyle X=(X)_{t<\xi }}$ is defined such that if ${\displaystyle t\to \xi }$, then ${\displaystyle \mathbb {P} }$-almost surely on ${\displaystyle \{\xi <\infty \}}$ we have ${\displaystyle X_{t}\to \infty }$ in the one-point compactification ${\displaystyle {\widehat {M}}:=M\cup \{\infty \}}$. In that case we extend the process path-wise by ${\displaystyle X_{t}:=\infty }$ for ${\displaystyle t\geq \xi }$.

Semimartingales on a manifold

A process ${\displaystyle X}$ is a semimartingale on ${\displaystyle M}$, if for every ${\displaystyle f\in C^{2}(M)}$ the random variable ${\displaystyle f(X)}$ is an ${\displaystyle \mathbb {R} }$-semimartingale, i.e. the composition of any smooth function ${\displaystyle f}$ with the process ${\displaystyle X}$ is a real-valued semimartingale. It can be shown that any ${\displaystyle M}$-semimartingale is a solution of a stochastic differential equation on ${\displaystyle M}$. If the semimartingale is only defined up to a finite lifetime ${\displaystyle \xi }$, we can always construct a semimartingale with infinite lifetime by a transformation of time. A semimartingale has a quadratic variation with respect to a section in the bundle of bilinear forms on ${\displaystyle TM}$.

Introducing the Stratonovich Integral of a differential form ${\displaystyle \alpha }$ along the semimartingale ${\displaystyle X}$ we can study the so called winding behaviour of ${\displaystyle X}$, i.e. a generalisation of the winding number.

Stratonovich integral of a 1-form

Let ${\displaystyle X}$ be an ${\displaystyle M}$-valued semimartingale and ${\displaystyle \alpha \in \Gamma (T^{*}M)}$ be a 1-form. We call the integral ${\displaystyle \int _{X}\alpha$ :=\int \alpha (\circ dX)} the Stratonovich integral of ${\displaystyle \alpha }$ along ${\displaystyle X}$. For ${\displaystyle f\in C^{\infty }(M)}$ we define ${\displaystyle f(X)\circ \alpha (\circ dX):=f(X)\circ d(\int _{X}\alpha )}$.

Definition

An SDE on ${\displaystyle M}$ is a pair ${\displaystyle (A,Z)}$, where

• ${\displaystyle Z=(Z_{t})_{t\in \mathbb {R} _{+}}}$ is a continuous semimartingale on a finite-dimensional ${\displaystyle \mathbb {R} }$-vector space ${\displaystyle E}$; and
• ${\displaystyle A:M\times E\to TM}$ is a (smooth) homomorphism of vector bundles over ${\displaystyle M}$

${\displaystyle A:(x,e)\mapsto A(x)e}$

where${\displaystyle A(x):E\to TM}$ is a linear map.

The stochastic differential equation ${\displaystyle (A,Z)}$ is denoted by

${\displaystyle dX=A(X)\circ dZ}$

or

${\displaystyle dX=\sum \limits _{i=1}^{r}A_{i}(X)\circ dZ^{i}.}$

The latter follows from setting ${\displaystyle A_{i}:=A(\cdot )e_{i}}$ with respect to a basis ${\displaystyle (e_{i})_{i=1,\dots ,r}}$ and ${\displaystyle \mathbb {R} }$-valued semimartingales ${\displaystyle (Z^{i})_{i=1,\dots ,r}}$ with ${\displaystyle Z=\sum \limits _{i=1}^{r}Z^{i}e_{i}}$.

As for given vector fields ${\displaystyle A_{1},\dots ,A_{r}\in \Gamma (TM)}$ there is exactly one bundle homomorphism ${\displaystyle A}$ such that ${\displaystyle A_{i}:=A(\cdot )e_{i}}$, our definition of an SDE on ${\displaystyle M}$ as ${\displaystyle (A_{1},\dots ,A_{r},Z)}$ is plausible.

If ${\displaystyle Z}$ has only finite life time, we can transform the time horizon into the infinite case.^[7]

Solutions

Let ${\displaystyle (A,Z)}$ be an SDE on ${\displaystyle M}$ and ${\displaystyle x_{0}:\Omega \to M}$ an ${\displaystyle {\mathcal {F}}_{0}}$-measurable random variable. Let ${\displaystyle (X_{t})_{t<\zeta }}$ be a continuous adapted ${\displaystyle M}$-valued process with life time ${\displaystyle \zeta }$ on the same probability space such as ${\displaystyle Z}$. Then ${\displaystyle (X_{t})_{t<\zeta }}$ is called a solution to the SDE

${\displaystyle dX=A(X)\circ dZ}$

with initial condition ${\displaystyle X_{0}=x_{0}}$ up to the life time ${\displaystyle \zeta }$, if for every test function ${\displaystyle f\in C_{c}^{\infty }(M)}$ the process ${\displaystyle f(X)}$ is an ${\displaystyle \mathbb {R} }$-valued semimartingale and for every stopping time ${\displaystyle \tau }$ with ${\displaystyle 0\leq \tau <\zeta }$, it holds ${\displaystyle \mathbb {P} }$-almost surely

${\displaystyle f(X_{\tau })=f(x_{0})+\int _{0}^{\tau }(df)_{X_{s}}A(X_{s})\circ \mathrm {d} Z_{s}}$,

where ${\displaystyle (df)_{X}:T_{x}M\to T_{f(x)}M}$ is the push-forward (or differential) at the point ${\displaystyle X}$. Following the idea from above, by definition ${\displaystyle f(X)}$ is a semimartingale for every test function ${\displaystyle f\in C_{c}^{\infty }(M)}$, so that ${\displaystyle X}$ is a semimartingale on ${\displaystyle M}$.

If the lifetime is maximal, i.e.

${\displaystyle \{\zeta <\infty \}\subset \left\{\lim \limits _{t\nearrow \zeta }X_{t}=\infty {\text{ in }}{\widehat {M}}\right\}}$

${\displaystyle \mathbb {P} }$-almost surely, we call this solution the maximal solution. The lifetime of a maximal solution ${\displaystyle X}$ can be extended to all of ${\displaystyle \mathbb {R} _{+}}$ , and after extending ${\displaystyle f}$ to the whole of ${\displaystyle {\widehat {M}}}$, the equation

${\displaystyle f(X_{t})=f(X_{0})+\int _{0}^{t}(df)_{X}A(X)\circ dZ,\quad t\geq 0}$,

holdsup to indistinguishability.^[8]

Remark

Let ${\displaystyle Z=(t,B)}$ with a ${\displaystyle d}$-dimensional Brownian motion ${\displaystyle B=(B_{1},\dots ,B_{d})}$, then we can show that every maximal solution starting in ${\displaystyle x_{0}}$ is a flow process to the operator

${\displaystyle L=A_{0}+{\frac {1}{2}}\sum \limits _{i=1}^{d}A_{i}^{2}}$.

Martingales with linear connection

To define martingales, we need a linear connection ${\displaystyle \nabla }$. Using the connection, we can characterise ${\displaystyle \nabla }$-martingales, if their anti-development is a local martingale. It is also possible to define ${\displaystyle \nabla }$-martingales without using the anti-development.

We write ${\displaystyle {\stackrel {m}{=}}}$ to indicate that equality holds modulo differentials of local martingales.

Let ${\displaystyle X}$ be an ${\displaystyle M}$-valued semimartingale. Then ${\displaystyle X}$ is a martingale or ${\displaystyle \nabla }$-martingale, if and only if for every ${\displaystyle f\in C^{\infty }(M)}$, it holds that

${\displaystyle d(f\circ X)\,\,{\stackrel {m}{=}}\,\,{\tfrac {1}{2}}(\nabla df)(dX,dX).}$

Brownian motion on a Riemannian manifold

Let ${\displaystyle (M,g)}$ be a Riemannian manifold with Laplace-Beltrami operator ${\displaystyle \Delta _{M}}$. An adapted ${\displaystyle M}$-valued process ${\displaystyle X}$ with maximal lifetime ${\displaystyle \xi }$ is called a Brownian motion${\displaystyle (M,g)}$, if for every ${\displaystyle f\in C^{\infty }(M)}$

${\displaystyle f(X)-{\frac {1}{2}}\int \Delta _{M}f(X)\mathrm {d} t}$

is a local ${\displaystyle \mathbb {R} }$-martingale with life time ${\displaystyle \xi }$. Hence, Brownian motion Bewegung is the diffusion process to ${\displaystyle {\tfrac {1}{2}}\Delta _{M}}$. Note that this characterisation does not provide a canonical procedure to define Brownian motion.