4 where B I denotes the operator B written in the interaction picture while D and D 1 are the memory function given by A new set of stochastic variables dv$_{S/E}$ and du$_{S/E}$ have been introduced through dξ$_{S/E}$ = dv$_{S/E}$ − idu$_{S/E}$ and dλ$_{S/E}$ = dv$_{S/E}$ + idu$_{S/E}$ , and verify Reporting the evolution of 〈 B ( t ) 〉$_{E}$ into the evolution of ρ$_{S}$ , a closed stochastic equation of motion for the system density is obtained: with By integrating out the evolution of the environment, a new stochastic term is found that depends not only on the noise at time t but also on its full history through the time integrals. Using second moments given by Eqs. (15) leads to: hermitian along a stochastic path. As a consequence, ex- pectation value of observables are complex loosing their physical meaning before averaging over the stochastic path. Nevertheless, we illustrate in the following that the new theory can be a very powerful tool. where Θ( t − t $^{′}$) = 1 if t > t ′ and 0 elsewhere. Inter- estingly enough, the stochastic equation given by (16) identifies with the stochastic master equation obtained in ref. [21] using a completely different method based on the Feynman-Vernon influence functional theory [49]. It should however be kept in mind that different strategies to design the stochastic equation (see discussion in sec- tion II A) would have given a different stochastic master equation. The Caldeira-Leggett (CL) model [37] corresponds to a single harmonic oscillator coupled to an environment of harmonic oscillators initially at thermal equilibrium, i.e. Despite the apparent complexity of Eq. (16), the QMC approach has been recently applied with success to the spin-boson model coupled to a heat bath of oscillators [43]. In particular, the introduction of (9) seems to cure the numerical difficulties that have been encountered in this model [46]. By projecting the environment effect onto the system density evolution we do not need any- more to follow the environment density and we expect that a rather limited number of trajectories will be suffi- cient to accurately simulate the onset of dissipation and fluctuation in an open quantum system. Eq. (16) is the equation that is solved in practice. It should be noted that the present stochastic process differs signif- icantly from conventional approaches. Indeed, accord- ing to the noise properties, system densities are non- and B ≡ − ∑ $_{n}$κ$_{n}$ x$_{n}$ [19]. Here, H$_{c}$ = Q $^{2}$∑ n κ 2 n 2 m n ω 2 n is the counter-term that insures that the physical frequency is ω 0 . In the following, ε is either +1 (harmonic case) or − 1 (inverted parabola case). Such a model can be solved exactly. As shown in ref. [43], the two functions D and D 1 defined by Eqs. (14) and estimated along the stochastic trajectories identify with the standard times correlation functions: where J ( ω ) ≡ ∑ n κ 2 n 2 m$_{n}$ω$_{n}$ δ ( ω − ω$_{n}$ ) denotes the spectral III. APPLICATION D ( t, s ) ≡ i 〈 [ B ( t ) , B ( t − s )] 〉$_{E}$ , and D 1 ( t, s ) ≡ 〈{ B ( t ) , B ( t − s ) } + 〉$_{E}$ − 2 〈 B ( t ) 〉$_{E}$ 〈 B ( t − s ) 〉$_{E}$ . (14) du$_{S}$du$_{E}$ = dv$_{S}$dv$_{E}$ = dt 2 $_{ℏ}$, du$_{S}$dv$_{E}$ = dv$_{S}$du$_{E}$ = 0 . (15) dρ$_{S}$ = dt i ℏ [ H$_{S}$, ρ$_{S}$ ] + dt [ Q, ρ$_{S}$ ] ∫ t 0 dsD ( t − s ) 〈 Q ( s ) 〉$_{S}$ + dξ ( t )[ Q, ρ$_{S}$ ] + dη ( t ) { Q − 〈 Q 〉$_{S}$ , ρ$_{S}$ } (16) dξ ( t ) = dt ∫ t 0 D 1 ( t − s ) dv$_{E}$ ( s ) − dt ∫ t 0 D ( t − s ) du$_{E}$ ( s ) − idv$_{S}$ ( t ) , dη ( t ) = du$_{S}$ ( t ) . (17) dη ( t ) dη ( t $^{′}$) = 0 , dξ ( t ) dη ( t $^{′}$) = − dt 2 $_{ℏ}$Θ( t − t $^{′}$) D ( t − t $^{′}$) , dξ ( t ) dξ ( t $^{′}$) = − idt 2 $_{ℏ}$D 1 ( | t − t $^{′}$| ) , H$_{S}$ = H$_{c}$ + P 2 2 M + ε 1 2 Mω 2 $_{0}$Q$^{2}$, (18) H$_{E}$ = ∑ n ( p 2 n 2 m$_{n}$ + 1 2 m$_{n}$ω 2 $_{n}$x 2 n ) (19) D ( τ ) = 2 ℏ ∫ + ∞ 0 dωJ ( ω ) sin( ωτ ) , (20) D 1 ( τ ) = 2 ℏ + ∞ 0 dωJ ( ω ) coth( ℏ ω/ 2 k$_{B}$T ) cos( ωτ ) , (21)