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.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?
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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