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Consider the following stochastic partial differential equation:

$$ t \frac{\partial^2}{\partial t^2}\Phi(x,t)=- \dot J x \frac{\partial}{\partial x}\Phi(x,t) $$

Where $\dot J$ is not "Gaussian noise," but is a different type of noise that is generated by adding random values that are $J$-distributed to the input data. The $J$-distribution is a close cousin of the K-distribution and is defined as:

$$J_t(x):=\exp\bigg(\frac{t}{\log x}\bigg)$$

For $x\in (0,1)$ and $t>0.$ The reason for choosing $\dot J$-noise is because it is compatible with the PDE already. What I mean by this is that $J_t(x)$ is a solution to the non-stochastic version of the PDE!

Therefore, adding $\dot J$-noise adds noise to the solution $J_t(x),$ making it not a smooth analytic function anymore but a stochastic version of the smooth analytic function.

Here is a plot of the $J$-distribution for several distinct values of $t:$

enter image description here

You can imagine with addition of the $\dot J$-noise the curves will look something like this (rotated accordingly):

enter image description here

How would you write down the noisy solution, that is, the stochastic version of $J_t(x)?$

Well, I think I need "multiplicative noise" as opposed to "additive noise" in order for the stochastic paths to remain inside $X=(0,1)^2.$ And I already know that this $J$-distribution behaves multiplicatively and not additively.

I searched for how to introduce stochasticity into the solution and found this link stochastic function. Here they just say $f(t)=L(t)+\epsilon(t).$ They add a noisy term $\epsilon$ to make the deterministic function stochastic.

Edit:

White noise can be written as an infinite sum over eigenfunctions with Gaussian coefficients. I want to replace these Gaussian coefficients with coefficients from the $J$-distribution.

J. Zimmerman
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  • If you haven’t encountered ”polynomial chaos expansions”, this might be of interest to you. It may be used to propagate uncertainty in a model by decomposing into an orthogonal basis depending on the distribution of the random variable. – AxelT Aug 03 '23 at 17:46
  • (a)"multiplicative stocastic heat equation", (b) "directed polymers in random media" – user619894 Aug 03 '23 at 18:35
  • I don't quite understand your question. In the original equation you wrote does the noise depend on both time/space $\dot{J}=\dot{J}(x,t)$? So you have $\dot{J}(x,t)=J_{t}(x)=exp(t/logx)$? – Thomas Kojar Dec 25 '23 at 06:05
  • You basically have a heat equation $$D(x,t)\partial_{t}\Phi=\partial_{xx}\Phi$$ for $D(x,t)=\frac{-t\dot{J}}{x}$ (by switching notation to make it more obvious). – Thomas Kojar Dec 25 '23 at 06:07
  • Or do you want $\dot{J}$ to be some type of distributional white noise? So the standard white noise is the derivative of Brownian motion, but you seem to want to take the "distributional derivative" of some different stochastic process, which one exactly? – Thomas Kojar Dec 25 '23 at 06:11
  • @ThomasKojar I know that I want the noise to have a pdf equal to that of the $J_t(x)$ distribution. In other words I want the values that the noise can take to be distributed according to the $J$-distribution. and I want to mimic additive gaussian white noise so that I need the random variables distributed according to $J_t(x)$ to be iid. – J. Zimmerman Dec 26 '23 at 23:49
  • What do you mean by "$J_{t}(x)$ distribution"? Above you write J_{t}(x) to be a deterministic function exp(...)? Can you carefully write the J-distribution? – Thomas Kojar Dec 26 '23 at 23:51
  • And then also the exact noise you are thinking about. The white noise is a distribution over L2-function https://math.stackexchange.com/questions/3753878/relation-between-the-white-noise-and-the-brownian-motion – Thomas Kojar Dec 26 '23 at 23:52
  • @ThomasKojar I could put the normalization constant in there to make $J$ a proper probability density function with parameter $t$ – J. Zimmerman Dec 26 '23 at 23:52
  • White noise can also be written as an infinite sum over eigenfunctions with Gaussian coefficients. So are you saying to replace the Gaussian coefficients by some J-distributed coefficients? Please include the details in your post on the construction of that noise to make it clear. – Thomas Kojar Dec 26 '23 at 23:53
  • @ThomasKojar Okay I'll add more details about the construction of the noise – J. Zimmerman Dec 26 '23 at 23:55
  • @ThomasKojar I made an edit saying I want the noise to replace Gaussian coefficients with J coefficients – J. Zimmerman Dec 27 '23 at 14:02

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