I'm looking for a stochastic interpretation of the heat equation for vector valued function.


The classical set up is the following : 
If $(M,g)$ is a riemannian manifold then we could consider the associated random walk, the Brownian motion with respect to $g$. This gives us a stochastic interpretation of the heat kernel $p(x,y,t)$ as the density of the probability (defined on the set of path coming from $x$) to join $y$ at times $t$. 


It is possible to extend this kind of idea to have a stochastic interpretation for heat kernel coming from fiber bundle ? I'm specially interested by the case of differentials forms on manifolds, where the fibers are endowed with the metric induced by $g$ on the forms, and the associated heat kernel. 


Thanks for helping.



Stochastic interpretation of heat kernel on fiber bundle
Let $P\longrightarrow M$ be a $G$ principal bundle endowed with a connection $1$-form $\omega$ (which has values in the Lie algebra $\mathfrak{g}$). If $X\_t^x$ denotes Brownian motion on $M$ starting at $x$ (or some other, let's say, diffusion process), then for each $u \in P\_x$, there exists a unique *lift* $U\_t^u$ of $X\_t$ starting at $u$, which is a process on $P$ with $\pi \circ U\_t = X\_t$ that is horizontal, meaning that
$$\int \omega(U\_t^u) \circ \mathrm{d} U^u\_t =0 ~~~~~\text{a.s.}$$


Let now in particular $\mathcal{V}$ be a vector bundle with connection $\nabla$ over $M$ and $P$ is its frame bundle, i.e. the fibers of which are
$$P\_x = \{A: \mathbb{R}^N \rightarrow \mathcal{V}\_x \mid A ~\text{is an isomorphism}\}$$
with the connection $1$-form $\omega$ induced by $\nabla$. If $p\_t^L(x, y)$ is the heat kernel of $L := - \mathrm{tr}\nabla^2$, then we have the formula
$$p\_t^L(x, y)= \mathbb{E}\bigl[ U^u\_0 (U^u\_t)^{-1} \mid X^x\_t = y\bigr],$$
$u \in P\_x$ is arbitrary. Hence the heat kernel for sections in $\mathcal{V}$ may be in some sense interpreted as the transition density on the frame bundle $P$.


\Edit: To expand a bit: One can define for a section $f$ the function $\hat{f}:P \longrightarrow \mathbb{R}^N$ by
$$ \hat{f}(u) = u^{-1} f(\pi(u)),$$
which is then $G$-invariant. Then
$$ \int\_M p\_t^L(x, y) f(y) \mathrm{d}y = \mathbb{E}\bigl[U\_0^u (U\_t^u)^{-1}f(X^x\_t)\bigr] = U\_0^u \mathbb{E}\bigl[\hat{f}(U^u\_t)\bigr],$$
so that the heat operator associated to $L$ can be directly interpreted as the transition density with respect to the process $U\_t$.


There are lots of references on this, e.g.
- Ichiro Shigekawa, On Stochastic Horizontal Lifts  

- Elton P. Hsu, Stochastic Analysis on Manifolds
- Hackenbroch, Thalmaier, Stochastische Analysis (in German)


