Continuum Limit of Forward Kolmogorov Equation Friday, March 06, 2015 2:04 PM Please note that one of the equations (for ordinary Brownian motion) in Problem 1 was corrected on Wednesday night. And actually I did mean for you to compute global drift and diffusivity in part b. Homework 2 due Tuesday, February 17 at 2 PM. Actually there's a better way to derive the discrete-time FKE than the method I gave last time; this method is more general. The better way to go is to use the law of total probability by partitioning on the random kick rather than the previous state : The ideas going forward from here are the same: AppSDE15 Page 1
Plug into the law of total probability expression; we get an alternative (equivalent) form of the discrete-time forward Kolmogorov equation: It's equivalent to the previous expression through the change of variable Now mapping to refer to an underlying continuous time trajectory, as we did at the end of the last lecture, this gets rewritten as: Write this as a discrete rate of change: AppSDE15 Page 2
Recall that depends on because So what happens as we take LHS is straightforward: For the RHS, we proceed in a manner that is appropriate for diffusion processes but not for jump processes. It relies on the size of the noise being concentrated at small values. The implication is that it allows us to justify a Taylor expansion for about in the integrand, because the integrand will be concentrated to to small values of. AppSDE15 Page 3
Therefore, whenever we can justify that (not true for jump processes, but follows from technical conditions for ``diffusion processes'') then the continuum limit of a discrete noise process has the following continuous-time forward Kolmogorov equation describing the evolution of the probability density for its state: where: is known as the (local) drift coefficient (vector) is known as the (local) diffusion coefficient (matrix) The forward Kolmogorov equation for a diffusion process (in continuous time ) is also known as a Fokker-Planck equation. It is the Eulerian (probability-based) counterpart to the stochastic differential equation representation for state trajectories. And in general, as we'll discuss later, the drift and diffusion coefficients can depend on state and time: But let's come back to our particular model and show that it does give rise to a diffusion process in the continuous time limit. Let's check first that the remainder term is negligible AppSDE15 Page 4
What are the (local) drift vector and diffusion matrix? This also follows by statistical isotropy of the model. So the Fokker- Planck equation for our simplest of Brownian motion is: with as the scalar diffusion coefficient. (The diffusion coefficient can be reduced to a scalar when the diffusion coefficient matrix is just a multiple of the identity.) This Fokker-Planck equation for our simplest model for Brownian motion is just the ordinary heat equation or ordinary diffusion equation. Let's see that this Fokker-Planck description is consistent with the Wiener process (stochastic differential equation) description for the continuous-time limit which we previously derived. Recall the Lagrangian (trajectory-based) description gave us: So we see, that if the initial condition is deterministic then: A key fact that we will use repeatedly is that if we have a collection of jointly Gaussian random variables and deterministic matrices AppSDE15 Page 5
and deterministic constant vector then: is also a Gaussian random variable and you simply need to compute its mean and covariance matrix by using the linearity and bilinearity properties of those statistical objects. (In other words, Gaussian random variables form a vector space.) By this property of deterministic linear transformations preserving Gaussianity, we have that since is Gaussian, then so is. On the other hand, if we solve the Fokker-Planck equation for this model by using Fourier transform: These are just ODEs; the initial conditions come from taking the initial PDF and Fourier transforming that: Solve the ODEs, and then take the inverse Fourier transform: AppSDE15 Page 6
(This explicit solution method of the Fokker-Planck equation by Fourier transform only works for constant coefficients and some very special other cases.) What is the relation and difference between the local drift vector and diffusion matrix and ``global'' drift vector and diffusion matrix? For our simplest Brownian motion model: The drift vector and diffusion coefficient are the same for both global and local versions, but that's just because the local drift and diffusion coefficients are constant (don't depend on state or time) in our simplest model for Brownian motion. AppSDE15 Page 7