# Fokker–Planck equation for very degenerate diffusion processes

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 $$p$$?