# Probobalistic interpretation of a PDE

1. Mar 1, 2013

### kai_sikorski

Consider the following PDE. A lot of this is from "Numerical Analysis of an Elliptic-Parabolic Partial Differential Equation" by J. Franklin and E. Rodemich.

$\frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -1$

With $|x|<1, |y| < \infty$ and we require T(1,y)=0 for y>0 and T(-1,y)=0 for y<0, and T(x,y) → 0 as |y| → ∞.

The solution T(x,y) is related to a randomly accelerated particle whose position ζ(t) satisfies the SDE ζ''(t) = w(t), where w(t) is white Gaussian noise. If the initial position and velocity are ζ(0) = x and ζ'(0)=y, where |x|<1, then T(x,y) is the expected value of the first time at which the sosition ζ(t) equals ±1.

I'm wondering if there is a similar probabilistic interpretation of a similar problem but with an x dependent forcing.

$\frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -G(x)$

Intuitively it seems that maybe I can integrate the right hand side along the path of the random particle, but I'm not sure if this is right. This seems similar in flavor to the Feynman-Kac theorem, but I haven't been able to find a formulation of that theorem that quite works for what I want.

2. Mar 2, 2013

### kai_sikorski

Think I found what I need in Grigorius stochastic calculus book, it looks like my intuition was right.