It is well known that if $X_t$ is a stochastic process that solves the SDE $$dX_t = \mu(X_t,t)\,\mathrm{d}t + \sigma(X_t,t)\,\mathrm{d}W_t,$$ with $W_t$ a Wiener process, then the associated probability density $p(x,t)$ solves $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \nabla \nabla : (Dp),$$ where $D_{ij} =\sum_{k} \sigma_{ik}\sigma_{jk}$ is symmetric, positive semi-definite. (as summarized e.g. here)

Now, in the same article, it is written that by interpreting the stochastic SDE in the Stratonovich sense rather than the Ito sense, one instead arrives at the corresponding equation $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \sum_{i,k} \frac{\partial}{\partial x_i}\left( \sigma_{ik} \sum_{j}\frac{\partial}{\partial x_j}\left[\sigma_{kj} p\right]\right).$$ I'm wondering if there is yet a third way to interpret the SDE which would lead to a parabolic equation in divergence-form for the probability density? I.e. to $$\frac{\partial p}{\partial t} = -\nabla \cdot (\mu p) + \frac 12 \sum_{i,j} \frac{\partial}{\partial x_i}\left( D_{ij} \frac{\partial p}{\partial x_j}\right).$$ The motivation for the last form is that it seems to arise very natural in physics. Does anyone know if that's possible? Thank you.

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    $\begingroup$ I'm interested to know what you mean by it arises naturally in physics. Is there a mathematical advantage to that; is it experimentally easier to measure or something understood; or does it happen to show up in a couple of famous equations? Can you give a some examples? $\endgroup$
    – stochastic
    Dec 12, 2017 at 16:48

2 Answers 2


The divergence-form of the Fokker-Planck equation arises if $\sigma$ is scalar and separable in space and time, $$\sigma_{ij}(x,t)=f(x)g(t)\delta_{ij}.$$ Define $\bar{p}=fp$, $d\bar{t}/dt=g^2$, $d\bar{x}_i/dx_i=1/f$, $\bar{\mu}_i=\mu_i/(fg^2)$, then the Stratonovich FP equation $$\frac{dp}{dt}=-\sum_{i}\frac{d}{dx_i}(\mu_i p)+\tfrac{1}{2}\sum_{ijk}\frac{d}{dx_{i}}\left(\sigma_{ik}\frac{d}{dx_j}(\sigma_{kj} p)\right)$$ transforms (upon multiplication of both sides by $f/g^2$) into $$\frac{d\bar{p}}{d\bar{t}}=-\sum_{i}\frac{d}{d\bar{x}_i}(\bar{\mu}_i\bar{p})+\tfrac{1}{2}\sum_{i}\frac{d^2}{d\bar{x}_i^2}\bar{p},$$ which has the desired divergence-form.

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    $\begingroup$ Only in dimension d=1.... $\endgroup$ Dec 11, 2017 at 22:21
  • $\begingroup$ indeed, or at least $\sigma$ should be a scalar --- thank you for correcting me! $\endgroup$ Dec 12, 2017 at 7:23

I think there is a thing called $A$-type stochastic differential equation that does that. See https://link.springer.com/content/pdf/10.1007%2Fs11467-017-0718-2.pdf


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