Beta-gamma random variables and intertwining relations between certain Markov processes
Philippe Carmona, Frédérique Petit, Marc Yor · Revista Matemática Iberoamericana · 1998
Proof.The result (2.a) is obtained by computing E f(X t+s )jG s ] in two dierent ways.On one hand, we have E f(X t+s )jG s ] = E E f(X t+s )jG t+s ]jG s ] = E f(Y t+s )jG s ] = Q t f(Y s ) : On the other hand, E f(X t+s )jG s ] = E E f(X t+s )jF s ]jG s ] = E P t f(X s )jG s ] = P t f(Y s ) :We now present six classes of examples of intertwining where the hypotheses made in Proposition 2.1 are in force. Dynkin's criterion.This is, undoubtedly, one of the best known, and oldest, examples of intertwining between two Markov processes (see 14]).Here, we s t a r t with a Markov process (Y t t 0) taking its values in a measurable space F Y is Markovian with respect to (G t ), with semi-group (Q t t 0).We assume that there exists a measurable application : F ;! E such that for every measurable function f : E ;! R + , the quantity Q t (f )(y) only depends, through y, o n (y) : Now, if x = (y), we dene P t f(x) = Q t (f )(y).It is now easy to see that the process (X t def = (Y t ) t 0) is Markovian with respect to (F t ) = ( G t ), and has semi-group (P t t 0).Moreover, by denition of (P t t 0), we have Q t = P t with f(y) = f((y))so that the hypotheses of Proposition 2.1 are satised.A particularly important example of this situation is obtained by taking Brownian motion in R n for (Y t t 0), and (X t = jY t j t 0),