mathematical-neuroscience.com The generating functional be an infinite dimensional generalization for the familiar generating function for a single random variable.

single random variable.

[PDF]Path Integral Methods for Stochastic Differential Equations

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by CC Chow - ‎2015

Feb 12, 2015 - closed form and must be tackled approximately using methods that include ... Although Wiener introduced path integrals to study stochastic ... agrams as a tool for carrying out perturbative expansions and introduces the “loop.

2 Moment Generating Functionals
The strategy of path integral methods is to derive a generating function or functional
for the moments and response functions for SDEs. The generating functional will
be an infinite dimensional generalization for the familiar generating function for a
single random variable. In this section we review moment generating functions and
show how they can be generalized to functional distributions.
Consider a probability density function (PDF) P(x) for a single real variable x.
The moments of the PDF are given by
xn = xnP(x)dx
and can be obtained directly by taking derivatives of the generating function
Z(J ) = eJx = eJxP(x)dx,
where J is a complex parameter, with
xn = 1
Z(0)
dn
dJn
Z(J )

J=0
.
Note that in explicitly including Z(0) we are allowing for the possibility that P(x)
is not normalized. This freedom will be convenient especially when we apply perturbation
theory. The generating function is



The analogy between stochastic systems and quantum theory, where path integrals
are commonly used, is seen by transforming the time coordinates in the path integrals
via t →
√
−1t . When the field ϕ is a function of a single variable t , then this would
be analogous to single particle quantum mechanics where the quantum amplitude can
be expressed in terms of a path integral over a configuration variable φ(t). When the
field is a function of two or more variables ϕ(
r, t), then this is analogous to quantum
field theory, where the



3 Application to SDE
Building on the previous section, here we derive a generating functional for SDEs.
Consider a Langevin equation,
dx
dt
= f (x, t)+g(x, t)η(t), (3)
on the domain t ∈ [0,T ] with initial condition x(t0) = y. The stochastic forcing term
η(t) obeys η(t) = 0 and η(t)η(t

) = δ(t −t

). Equation (3) is to be interpreted as
the Ito stochastic differential equation
dx = f (xt , t)dt + g(xt , t)dWt , (4)
where Wt is a Wiener process (i.e. Gaussian white noise), and xt is the value of x
at time t . We show how to generalize to other stochastic processes later. Functions
f and g are assumed to obey all the properties required for an Ito SDE to be well
posed [31]. In particular, the stochastic increment dWt does not depend on f (xt , t)
or g(xt , t) (i.e. xt is adapted to the filtration generated by the noise). The choice
between Ito and Stratonovich conventions amounts to a choice of the measure for
the path integrals, which will be manifested in a condition on the linear response or
“propagator” that we introduce below.


生成方程组

来自: cmp(const void*, const void*) 2014-03-22 11:24:32

举个例子, 在 3x3 的正方格点上, 每个格点都对应一个变量 θ_0 .. θ_8, 如下图

3x3 的正方晶格

每个格点都和它周围的四个格点有关系. 用了周期性边界条件, 最左边/上面的会和最右边/下面的连上.

然后我希望对每个格点 i, 生成一个方程. 例如, 对于 θ_0, 我想生成

θ_0 + θ_1 + θ_3 + θ_6 + θ_2 == 0

我对每个 i 都要生成一个长得一样的方程. 然后我要把它们联立成方程组, 求数值解.

请问这应该怎么做?

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