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.(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -1[/itex]

With [itex]|x|<1, |y| < \infty[/itex] 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.

[itex]\frac{1}{2} \frac{\partial^2 T}{\partial y^2} + y \frac{\partial T}{\partial x} = -G(x)[/itex]

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.

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# Probobalistic interpretation of a PDE

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