In this talk I shall start with a brief discussion of three motivating examples/questions. (1) Given a one-parameter group of unitary operators $( U_x )$ on a separable Hilbert space $\mathsf{h}$, set $V_t = U_{B(t)}$ ($t \geqslant 0$) where $B$ is the canonical Brownian motion. Thus $( V_t )$ is a unitary operator valued stochastic process. The semigroup of time-shifts on paths induces a semigroup of completely positive maps $( \sigma_t )$ on the algebra $L^\infty( \Omega; B(\mathsf{h}) ) \subset B(\mathsf{h}) \overline{\otimes} B( L^2( \Omega ) )$ with respect to which $V$ is easily seen to enjoy the cocycle relation $V_{r+t} = V_r \sigma_r( V_t )$ ($r,t \geqslant 0$). W. Arveson posed the questions, where does this structure arise beyond that of suitably randomised unitary evolutions?, and how can one classify the possibilities? (2) In mathematical physics, a quantum dynamical semigroup is a one-parameter semigroup of completely positive unital maps on an operator algebra of observables associated with a quantum system undergoing dissipation. How may such a semigroup be viewed as the Markov semigroup of some quantum Markov process? (3) What is a good ‘quantum’ analogue of Lévy processes on a Lie group $G$? Can QSA realise such processes on a compact quantum group $\mathbb{G}$ (in the sense of Woronowicz)?

In the rest of the first part of the talk I shall outline the basic framework of QSA, connecting it to elements of the Malliavin calculus, and relate it to examples (1) and (2) through respective concepts of quantum stochastic cocycle. For a clue as to what might be involved, imagine taking the canonical commutation relations of quantum mechanics $[p,q] = - \mathrm{i}$ (in units for which Plank’s constant equals $2 \pi$), and replacing $q$ by a position Brownian motion $( Q_t )$ and $p$ by a momentum Brownian motion $( P_t )$, and getting a quantum Brownian motion $( Q_t , P_t )$ satisfying $[ P_t, Q_s ] = - \mathrm{i} \min{s,t}$.

After the break I intend (time permitting) to (a) address Questions (3), and/or (b) discuss a quadratic form approach to the construction of quantum stochastic cocycles analogous to the Stroock–Varadhan martingale problem approach to classical diffusion processes which has revolutionised the Ito–McKean canon. Both of these are areas of current research.