Suppose we have a Brownian motion(or transition density) on a Lie group G and a Riemmanian manifold H on which G acts transitively and isometrically. Can we construct a Brownian motion( or transition density) on H?
Any help will be useful.



Brownian motion on Homogeneous spaces
This question is not as trivial as some people think. 


One point is that a quotient of a Markov chain is not Markov in general; there is a certain condition that has to be satisfied. More precisely, if $\pi\_x$ are the transition probabilities of a Markov chain on a space $X$, and $X\to \overline X,\; x\mapsto \overline x$ is a quotient map, then the quotient chain on $\overline X$ is Markov if and only if the images $\overline{\pi\_x}$ and $\overline{\pi\_y}$ of transition probabilities $\pi\_x$ and $\pi\_y$ coincide whenever $\overline x = \overline y$. 


The other point is that in the presence of a group of symmetries it is natural to require that the transition probabilities of the quotient chain be invariant with respect to this group. 


In your situation the quotient manifold (which I denote by $X$ instead of $H$ in order to avoid confusion with subgroups) is $X=G/K$, where $K$ is a compact subgroup of $G$ (because $G$ acts by isometries, so that point stabilizers are compact). The most standard example is, of course, provided by non-compact Riemannian symmetric spaces.


Assuming the original chain (or process) on $G$ is space homogeneous, the transition probabilities on $G$ are translates of a single probability measure $\mu$:
$$
\pi\_g = g\pi\_e = g\mu \;. 
$$
Then the above conditions amount to bi-$K$-invariance of the measure $\mu$. Namely, this is a necessary and sufficient condition for the quotient chain on $X$ to be (i) Markov, and (ii) $G$-invariant.


