Eulerian (Probability-Based) Approach

Transcription

1 Eulerian (Probability-Based) Approach Tuesday, March 03, :59 PM Office hours for Wednesday, March 4 shifted to 5:30-6:30 PM. Homework 2 posted, due Tuesday, March 17 at 2 PM. correction: the drifts and diffusivities you should compute are local, not "global" Reading: Gardiner, Secs. 1.2, 3.8.1, Einstein (optional) To prepare for the Eulerian (probability-based) approach to our simplest model for Brownian motion, we need to adapt the law of total probability to the case where one works with continuous random variables. Continuing from last time, if the cases on which the law of total probability is conditioned refers to the possible values which a discrete random variable can take, then the law of total probability can be written: where is the state space of. This can be extended (after quite a bit of technical work) to continuous random variables by the obvious changes of replacing probabilities by probability densities and sums by integration: is the conditional probability density of given that is near. It is a PDF w.r.t. the variable. The variable acts as a parameter, telling how the probability density for should be modified. Now we're ready to formulate the Eulerian (probability-based) approach to the simplest for Brownian motion. The first step is to express the dynamics in terms of a forward Kolmogorov equation which tells you how to update probabilities, as opposed to updating states. Lagrangian (trajectory-based) equation for updating states is: with prescribed iid random noise. Now we reformulate this equation in terms of updating probabilities. We begin by applying the law of total probability to the PDF for the next state given the information about the current state. (This is the standard approach for discrete-time stochastic processes that have the Markov property, meaning that the random updates to the state of the system only depend on the current state and some AppSDE15 Page 1

2 independent noise.) The dynamics will be encoded entirely in the conditional probability density for the next state given the current state This conditional probability is often called a transition probability density. Probabilities can be more reliably computed than probability densities, so let's recognize that:. We'll compute the left hand side because it will be more secure to work with probabilities here than with probability densities directly. There are now two routes to simplification: the unknown part (to the left of the bar) refers to some known information (to the right of the bar) the random variable is independent of all other random variables appearing in the expression. First we exploit the known information: Simply plugging in the known information into the expression to the left of the bar can be justified when the expression on the left is a continuous function of that information. Otherwise, have to worry a bit You should not trash the condition after you plug it into the expression to the left of the bar. This is justified by the independence of the random variable (which only depends on.) from AppSDE15 Page 2

3 The red equality holds for any reasonable Borel subset. Since the integrals of the functions are the same for any reasonable (Borel) subset, the functions themselves must be the same: So substituting this into the law of total probability, we obtain the discrete-time forward Kolmogorov equation (In deriving the discrete-time forward Kolmogorov equation on the homework, you won't get credit for simply literally repeating these steps and deriving this exact equation. You should express your forward Kolmogorov equation in terms of the random variables for speed and direction.) The forward Kolmogorov equation expresses mathematically how to update the PDF for the current state to the PDF for the next state, using the known PDF prescribed in the model. AppSDE15 Page 3

4 The reason it appears as a convolution of the current PDF with the noise PDF is because those are independent random variables added together. Such equations are usefully analyzed and solved using (Fourier) transform, which converts convolution to multiplication. In probability language, the Fourier transform of a probability density is known as the characteristic function of the random variable. This process will allow an exact solution to the model, but this is not our main focus. As with any dynamical process, the updating rule has to be supplemented with some initial condition. For the trajectory-based (Lagrangian) point of view,that just meant specifying the initial state possibly with some probability distribution. In probability-based (Eulerian) point of view, the initial condition just is the initial PDF. What if the initial state is deterministic, i.e.,, a deterministic state. This can be encoded as a ``generalized'' PDF to allow Dirac delta function: What does this mean? A Dirac delta function is defined as an object (functional or generalized function, not a function) which has the property that: for any bounded continuous function. Now we will contemplate taking a continuous time limit of the simplest model for Brownian motion, but now in an Eulerian (probability-based) framework. As before, we make the link to a continuous-time trajectory via. The translation to the PDFs will be: (just a notation change, putting time argument as explicit parameter) Substituting this relationship into the discrete-time FKE: AppSDE15 Page 4

5 And just refer to the current time as Now we study the depends on because. limit of this equation, remembering that To get a meaningful continuum equation, don't just directly take the limit; rather express the equation in terms of a discrete rate of change from the current state, then take the limit. AppSDE15 Page 5