Prove the density of this SDE is not smooth in a parameter

Question

Consider the following, 1-dimensional, equation

$$X_t^x = x + \int_0^t \mathbb{E} |X_s^x| \, ds + B_t , $$

where $B$ is a Brownian motion. This a McKean-Vlasov equation, sometimes called a nonlinear diffusion or mean-field equation. I can prove that it has a unique strong solution. What I would like to show is that its density is not smooth, i.e. $C^{\infty}$, in $x$. Or alternatively, I would like to show that for fixed $t$, the map $$ x \mapsto \mathbb{E} |X_t^x| $$ is not smooth.

Some McKean-Vlasov SDEs can be solved explicitly, but in this case the presence of the modulus in the coefficients makes it hard to find an explicit solution. I therefore do not know the density or an expression for $\mathbb{E} |X_t^x|$ explicitly.

• Without knowing the density or an expression for $\mathbb{E} |X_t^x|$ explicitly, how can I show that they are not smooth?

Answer 1

1. the $X_t^x$ are normal
2. With $\mu(x,t) = \mathbb E X_t^x$ I think that I can show that if $\mu(0,t) = at + O(t^2)$ then $\mu(0,t) \sim t^{\frac 32}$
3. Proof: $\mathbb E |X_t^x| = \mu(x,t) + 2\sqrt{s} \phi(\frac {\mu} {\sqrt s})$ so from defining equation $\mu(0,t) = \int^t \mu(0,t) + + 2\sqrt{s} \phi(\frac {\mu} {\sqrt s}) ds$ $\approx \int^t as + 2\sqrt s \phi(0) ds$