Author:
Gu Yu,Henderson Christopher
Abstract
Abstract
For a Brownian directed polymer in a Gaussian random environment, with q(t, ⋅) denoting the quenched endpoint density and
Q
n
(
t
,
x
1
,
…
,
x
n
)
=
E
[
q
(
t
,
x
1
)
…
q
(
t
,
x
n
)
]
,
we derive a hierarchical PDE system satisfied by
{
Q
n
}
n
⩾
1
. We present two applications of the system: (i) we compute the generator of
{
μ
t
(
d
x
)
=
q
(
t
,
x
)
d
x
}
t
⩾
0
for some special functionals, where
{
μ
t
(
d
x
)
}
t
⩾
0
is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with d ⩾ 3, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive O(t
2/3) scaling.
Subject
Applied Mathematics,General Physics and Astronomy,Mathematical Physics,Statistical and Nonlinear Physics
Cited by
2 articles.
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