Consider a diffusion process

$$X_t=X_0+\int_0^t {\bf 1}_{\{X_s>0\}}b(s,X_s)ds+ \int_0^t {\bf 1}_{\{X_s>0\}} a(s,X_s)dW_s,\quad \forall t\ge 0,$$

where $a: \mathbb R_+\times \mathbb R\to [1,2]$ and $b:\mathbb R_+\times \mathbb R\to\mathbb R_+$ are regular functions. Assume $X_0>0$ admits a density function $p_0:(0,\infty)\to\mathbb R_+$, then $X_t\ge 0$ and $0$ is an absorption point. Denote by $\mu_t$ the distribution of $X_t$. It is known (e.g. On the marginal distributions of an absorbed diffusion) that $\mu_t$ is a linear combination of a Dirac measure and a sub-probability with density, i.e.

$$\mu_t(dx) = \alpha(t)\delta_{0}(dx) + p(t,x)dx,$$

where $\alpha(t)\ge 0$ and $p(t,x)\ge 0$ satisfy

$$\alpha(t) + \int_0^{\infty}p(t,x)dx =1.$$

Does there exist a PDE of Fokker–Planck type for $p$?

Any answer, references or comments are appreciated.


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