Dynamic inflation with quantum fluctuations can be modeled as diffusion process in phase space. The Fokker-Planck (FP) equation describes the evolution of probability P(,t): P(,t)t=[V()3HP(,t)]Drift+22[H382P(,t)]Diffusion, tP(,t)=Drift3HV()P(,t)]+Diffusion22[8p2H3P(,t)],
where:
Drift Tribe: Describes the inflaton motion due to potential V() V().
Diffusion Tribe: Represents quantum fluctuations H/2 dH/2p.
2. Steady-State Solution for P() P() In condition slow-roll (P/t0
P/t0), the steady-state solution is: P()exp(82V()3H4). P()exp(3H48p2V()).
Derivative: Integrate the steady-state FP equation: V()3HP()=H382dP()d. 3HV()P()=8p2H3ddP().
The exponential solution is obtained by separating the variables: dPP=82V()3H4dP()e82V()3H4. PdP=3H48p2V()dP()and3H48p2V().
3. Transformation to P(L)P(L Residual vacuum energy LL related to the final inflaton value : LV().V().
By transformation of random variables , the probability distribution becomes: P()=P()ddexp(823H4)1V(). P(L)=P()dLdexp(3H48p2L)V()1.
For potential KKLT V()V0Aea V()V0Butand , derivative V()aAea V()aAeand , so that:P()exp(823H4)1V0. P()exp(3H48p2L)V0-L1
