524
A. Doikou et al.
fields with components x j , b j , respectively, and T denotes usual transposition. The
operator ˆ
L 0 is not in general self-adjoint (Hermitian), therefore we also introduce
the adjoint operator defined for any suitable function f (x, t) as
ˆ
L
†
0 f (x, t) =
1
2
M
i,j =1
∂ 2
∂x i ∂x j
g ij (x)f (x, t)
−
M
j =1
∂
∂x j
b j (x)f (x, t)
. (3)
Then, two distinct time evolution equations emerge.
1. The Fokker–Planck equation:
∂ t 1 f (x, t 1 ) = ˆ
L
†
0 f (x, t 1 )
(4)
t 1 ≥ t 2 , with known initial condition f (x, t 2 ) = f 0 (x).
2. The Kolmogorov backward equation:
− ∂ t 2 f (x, t 2 ) = ˆ
L 0 f (x, t 2 )
(5)
t 2 ≤ t 1 , with known final condition f (x, t 1 ) = f f (x).
To simplify the complicated situation of non-constant diffusion coefficients we
employ a local change of frame, which reduces the ˆ
L operator to the simpler form
with constant diffusion coefficients, and an effective drift. We then compute the
generic path integral by requiring that the fields involved satisfy discrete time
analogues of stochastic differential equations (SDEs). These equations naturally
emerge when computing the path integral via the time discretization scheme. This
leads to the computation of the measure, which turns out to be an infinite product of
Gaussians.
Our second objective is to explore links between SDEs, and quantum integrable
systems. To illustrate these associations we discuss a typical exactly solvable
discrete quantum system, the discrete non-linear Schrödinger hierarchy. We express
the quantum integrals of motion as second order differential operators after a
suitable rescaling of the fields and we then extract a hierarchy of associated SDEs,
which can be in principle solved by means of suitable integrator factors. The first
two non-trivial members of the hierarchy correspond to the discrete stochastic
transport and heat equations. The discrete stochastic Burgers equation is also
obtained from the discrete stochastic heat equation through the analogue of the
Cole–Hopf transformation (see also relevant [1]). More details on the derivation
of the reported results can be found in [2].
A. Doikou et al.
fields with components x j , b j , respectively, and T denotes usual transposition. The
operator ˆ
L 0 is not in general self-adjoint (Hermitian), therefore we also introduce
the adjoint operator defined for any suitable function f (x, t) as
ˆ
L
†
0 f (x, t) =
1
2
M
i,j =1
∂ 2
∂x i ∂x j
g ij (x)f (x, t)
−
M
j =1
∂
∂x j
b j (x)f (x, t)
. (3)
Then, two distinct time evolution equations emerge.
1. The Fokker–Planck equation:
∂ t 1 f (x, t 1 ) = ˆ
L
†
0 f (x, t 1 )
(4)
t 1 ≥ t 2 , with known initial condition f (x, t 2 ) = f 0 (x).
2. The Kolmogorov backward equation:
− ∂ t 2 f (x, t 2 ) = ˆ
L 0 f (x, t 2 )
(5)
t 2 ≤ t 1 , with known final condition f (x, t 1 ) = f f (x).
To simplify the complicated situation of non-constant diffusion coefficients we
employ a local change of frame, which reduces the ˆ
L operator to the simpler form
with constant diffusion coefficients, and an effective drift. We then compute the
generic path integral by requiring that the fields involved satisfy discrete time
analogues of stochastic differential equations (SDEs). These equations naturally
emerge when computing the path integral via the time discretization scheme. This
leads to the computation of the measure, which turns out to be an infinite product of
Gaussians.
Our second objective is to explore links between SDEs, and quantum integrable
systems. To illustrate these associations we discuss a typical exactly solvable
discrete quantum system, the discrete non-linear Schrödinger hierarchy. We express
the quantum integrals of motion as second order differential operators after a
suitable rescaling of the fields and we then extract a hierarchy of associated SDEs,
which can be in principle solved by means of suitable integrator factors. The first
two non-trivial members of the hierarchy correspond to the discrete stochastic
transport and heat equations. The discrete stochastic Burgers equation is also
obtained from the discrete stochastic heat equation through the analogue of the
Cole–Hopf transformation (see also relevant [1]). More details on the derivation
of the reported results can be found in [2].