In this chapter, we apply the path integral method to stochastic differential equations (SDEs) and obtain the same results as those for the asymmetric random walk, thereby proving the equivalence between the two models.
The idea of using path integrals to study stochastic differential equations has a long history. In his work [5], Feynman already established the relationship between path integrals and linear stochastic differential equations. For the nonlinear case, there have been many studies, but they are somewhat confusing; for instance, reference [8] even provided incorrect results.
Starting from the concept of discretization in path integrals, this article explicitly establishes the Jacobian relationship between the two path integral measures, thereby constructing the path integral for nonlinear stochastic differential equations as well. The results of this article are consistent with those in reference [9].
Concepts
The focus of this study is limited to stochastic ordinary differential equations. The difference between these and general ordinary differential equations lies in the introduction of a Brownian motion term. A common class of stochastic differential equations is: dx(t) = p(x(t), t)dt + \sqrt{\alpha} dW_t. \tag{48} Where W_t represents a standard Brownian motion. Due to the introduction of the stochastic term, the solution x(t) is no longer deterministic but follows a certain probability distribution.
In the field of stochastic differential equations, there are many quantities of interest, such as the expectation, variance, or stability of a certain variable x. Various analytical techniques exist in the field of SDEs, but ideally, the most straightforward approach is to directly find the probability distribution of x(t) and then study it. Path integrals provide exactly such a method for finding the probability distribution.
Linear Stochastic Differential Equations
We begin with the following linear stochastic differential equation: dx(t) = x(t)dt + \sqrt{\alpha} dW_t, \tag{49} We wish to solve this equation. Our focus is on the probability distribution of x; therefore, the goal is to find the probability of starting from x_a at time t_a and arriving at x_b at time t_b, which is the propagator (Green’s function). To construct the path integral, we need to find the probability functional P[x(t)] for reaching (t_b, x_b) from (t_a, x_a) via the path x(t).
We can start from the probability functional of Brownian motion. Since x(t) can be uniquely calculated given W(t), the probability of a given x(t) is the same as the probability of a given W(t), i.e., P[x(t)]\mathscr{D}x(t) = P[W(t)]\mathscr{D}W(t). \tag{50} We already know that the probability density functional for W(t) in Brownian motion is: P[W(t)] = \exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2(t)dt\right). \tag{51} Note that we still write this in the form of a derivative; this is merely formal (the derivative terms in the definition of path integrals are inherently formal), serving as a notation that is easy to understand and remember. Now we have: \begin{aligned} P[x(t)]\mathscr{D}x(t) &= P[W(t)]\mathscr{D}W(t) \\ &= \exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2(t)dt\right)\mathscr{D}W(t) \\ &= \exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}W(t) \end{aligned}, \tag{52} Up to this point, what we have done is easy to understand—it is simply a change of variables. Now, the final step is missing: the relationship between \mathscr{D}x(t) and \mathscr{D}W(t). Since \mathscr{D}x(t) and \mathscr{D}W(t) are defined as the limits of (n-1)-fold integrals, and the change of variables in finite-dimensional integrals involves a Jacobian determinant, the relationship between these two should also involve a Jacobian: \mathscr{D}W(t) = \mathcal{J}[x(t)]\mathscr{D}x(t), \tag{53} Due to the infinite nature of \mathscr{D}, the Jacobian here is the determinant of an \infty \times \infty square matrix.
Notice that the relationship between W(t) and x(t) is linear. We know that the Jacobian of a linear transformation is a constant, and this case is no exception. Therefore, we can ignore this constant for now and determine it later through normalization after obtaining the result. Up to a constant factor, we have: \begin{aligned} P[x(t)]\mathscr{D}x(t) &= \exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}W(t) \\ &= \exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right)\mathscr{D}x(t) \end{aligned}, \tag{54} Which means: P[x(t)] = \exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}(t)-x(t)\right]^2 dt\right). \tag{55} With this result, we can use the path integral to calculate the propagator: \begin{aligned} \int P[x(t)]\mathscr{D}x(t) &= \int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)-2\dot{x}(t) x(t)\right] dt\right)\mathscr{D}x(t) \\ &= \exp\left(\frac{1}{2\alpha}\left(x_b^2-x_a^2\right)\right) \times \\ & \qquad \int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)\right] dt\right)\mathscr{D}x(t) \end{aligned}. \tag{56} The path integral part: K(x_b,t_b;x_a,t_a) = \int\exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}^2(t)+x^2(t)\right] dt\right)\mathscr{D}x(t) \tag{57} is a basic Gaussian path integral, which we can solve exactly.
Calculating the Jacobian
Now we derive the path integral for the general form of the stochastic differential equation: dx(t) = p(x(t), t)dt + \sqrt{\alpha} dW_t, \tag{58} As in the previous section, it is easy to obtain: \begin{aligned} P[x(t)]\mathscr{D}x(t) &= P[W(t)]\mathscr{D}W(t) \\ &= \exp\left(-\int_{t_a}^{t_b} \frac{1}{2}\dot{W}^2 dt\right)\mathscr{D}W(t) \\ &= \exp\left(-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}-p(x,t)\right]^2 dt\right)\mathscr{D}W(t) \end{aligned}. \tag{59}
The key is the relationship between \mathscr{D}x(t) and \mathscr{D}W(t). In the linear case, we knew it was a constant and could ignore it until normalization; however, in the nonlinear case, we cannot bypass it and must find a way to calculate its determinant.
Discretize the equation as: (x_k - x_{k-1}) - \frac{1}{2}\epsilon\left[p(x_k,t_k)+p(x_{k-1},t_{k-1})\right] = \sqrt{\alpha} (W_k-W_{k-1}), \tag{60} where k=1,2,\dots,n, and W_0\equiv W(t_a), W_n\equiv W(t_b), x_0\equiv x(t_a), x_n\equiv x(t_b) are pre-given initial and boundary conditions, \epsilon=\frac{t_b-t_a}{n} is the time interval, and x_k\equiv x(t_a + k\epsilon). Note that we took the average for p(x,t). This is necessary; failing to take the average would lead to incorrect results. Because \frac{x(t)-x(t-\epsilon)}{\epsilon} as an approximation of \dot{x}(t) only has zero-order accuracy, whereas it is actually closest to \dot{x}(t-\epsilon/2), where it has first-order accuracy in \epsilon. That is to say, when we write (x_k-x_{k-1})/\epsilon, it actually represents the derivative at the midpoint between x_k and x_{k-1}. Therefore, the discretization of p(x,t) must also take the midpoint part. We can use \frac{1}{2}\left[p(x_k,t_k)+p(x_{k-1},t_{k-1})\right] or p\left(\frac{x_k+x_{k-1}}{2},\frac{t_k+t_{k-1}}{2}\right); both have equivalent accuracy and yield the same result (in integral approximation, we can use the trapezoidal rule or the midpoint rule to approximate the area, which carries a similar meaning).
We write: \left\{\begin{aligned} &\mathscr{D}x(t) \approx dx_1 dx_2 \dots dx_{n-1} \\ &\mathscr{D}W(t) \approx dW_1 dW_2 \dots dW_{n-1} \end{aligned}\right. . \tag{61} Then, using finite-dimensional calculus, we find the (approximate) Jacobian determinant. Note that the Jacobian matrix from W_k to x_k is a triangular matrix; thus, the determinant is the product of the diagonal elements. The diagonal elements are: \frac{\partial W_k}{\partial x_k} = \frac{1}{\sqrt{\alpha}}\left(1-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right) \approx \frac{1}{\sqrt{\alpha}}\exp\left(-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right), \tag{62} Therefore, up to a constant factor, we have: \mathcal{J}[x(t)] \approx \prod_{k=1}^{n-1} \left(1-\frac{1}{2}\epsilon\frac{\partial p(x_k,t_k)}{\partial x_k}\right) \approx \exp\left[-\frac{1}{2}\left(\sum_{k=1}^{n-1}\frac{\partial p(x_k,t_k)}{\partial x_k}\right)\epsilon\right], \tag{63} Taking the limit n \to \infty, we have: \mathcal{J}[x(t)] = \exp\left(-\frac{1}{2}\int_{t_a}^{t_b}\frac{\partial p(x,t)}{\partial x}dt\right), \tag{64} Substituting this back, we get: \begin{aligned} P[x(t)]\mathscr{D}x(t) &= \exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\dot{x}-p(x,t)\right]^2 dt\right\} \exp\left(-\frac{1}{2}\int_{t_a}^{t_b}\frac{\partial p(x,t)}{\partial x}dt\right)\mathscr{D}x(t) \\ &= \exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\left(\dot{x}-p(x,t)\right)^2 + \alpha\frac{\partial p(x,t)}{\partial x} \right]dt\right\}\mathscr{D}x(t) \end{aligned}, \tag{65} Thus: \begin{aligned} P[x(t)] &= \exp\left\{-\frac{1}{2\alpha}\int_{t_a}^{t_b} \left[\left(\dot{x}-p(x,t)\right)^2 + \alpha\frac{\partial p(x,t)}{\partial x} \right]dt\right\} \\ &= \exp\left\{-\frac{1}{\alpha}\int_{t_a}^{t_b}\left[\frac{1}{2}\dot{x}^2-\dot{x}p(x,t)+\frac{1}{2}p^2(x,t)+\frac{1}{2}\alpha\frac{\partial p(x,t)}{\partial x}\right]dt\right\} \end{aligned}. \tag{66} Note that: \begin{aligned} \frac{d}{dt}\int p(x,t)dx &= \dot{x}\frac{\partial}{\partial x}\int p(x,t)dx + \frac{\partial}{\partial t}\int p(x,t)dx \\ &= \dot{x}p(x,t) + \int \frac{\partial p(x,t)}{\partial t}dx \end{aligned}, \tag{67} Using this identity, the above expression can be rewritten as: P[x(t)] = \exp\left[-\frac{1}{\alpha}\int_{t_a}^{t_b}\left(\frac{1}{2}\dot{x}^2-V(x,t)\right)dt\right]\exp\left(\frac{1}{\alpha}\int_{x_a}^{x_b}p(x,t)dx\right). \tag{68} Where: V(x,t) = -\frac{1}{2}\left(\alpha\frac{\partial p}{\partial x}+p^2\right) - \int \frac{\partial p}{\partial t}dx. \tag{69} Comparing this with the results of equations (16) \sim (19), we find that the result for the stochastic differential equation (48) is exactly the same as that for the asymmetric random walk.
Path Integral Method
We have now proven the equivalence between stochastic differential equations and the asymmetric random walk model. In the first chapter, we analyzed the partial differential equation satisfied by the asymmetric random walk model, which means we can immediately write down the partial differential equation corresponding to the stochastic differential equation. Thus, we have achieved the mutual transformation between the three, with the path integral method serving as the link. This also demonstrates the power of the path integral method.
Original address: https://kexue.fm/archives/3762
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