Let $b\in \mathbb R$ and $\sigma>0$ be given. For a fixed probability distribution $\mu_0$ on $\mathbb R$ s.t.

$$\int_{(0,\infty)}\mu_0(dx)=1,$$

consider the mean field SDE :

$$dX_t = \mathbf{1}_{\{X_t>0\}} \left[ bdt + \frac{\sigma}{1 + m_t {\bf 1}_{\{b>0\}}} dW_t \right],\quad \mbox{for all } t\ge 0,~~~~~~~~~~~~~~~~(\ast)$$

where $X_0\sim \mu_0$ is independent of the Brownian motion $(W_t)_{t\ge 0}$ and

$$m_t:=\int_{(0,\infty)}\mu_t(dx),\quad \mbox{for all } t\ge 0.$$

How can we show the existence and uniqueness of the (weak) solution to $(\ast)$?

Any answers, remarks or references are highly appreciated!

**REMARK :**

The case for $b\le 0$ is trivial. Indeed, $(\ast)$ reduces to $dX_t = \mathbf{1}_{\{X_t>0\}} \big[ bdt + \sigma dW_t \big]$ and the solution is given as $X_t=Y_{t\wedge \tau}$, where $Y_t:=X_0+bt+\sigma W_t$ and $\tau:=\inf\{t\ge 0: Y_t\le 0\}$. For the case $b>0$, $(\ast)$ turns to be

$$dX_t = \mathbf{1}_{\{X_t>0\}} \left[ bdt + \frac{\sigma}{1 + m_t} dW_t \right].$$

I do not find any literature on the existence of its solution.

**REMARK 2 :**

A heuristic argument is as follows : Let $\ell:\mathbb R_+\to [0,1]$ be some "nice" function. Consider the process

$$Y^{\ell}_t: = X_0+ bt+\int_0^t\frac{\sigma}{1+\ell(s)}dW_s.$$

Then we have $(Y^{\ell}_{t\wedge \tau^{\ell}})_{t\ge 0}$ is a solution to $(\ast)$ if $\mathbb P[\tau^{\ell}>t]=m_t$ for all $t\ge 0$, where $\tau^{\ell}:=\inf\{t\ge 0: Y^{\ell}_t\le 0\}$. Thus it remains to calculate the probability $\mathbb P[\tau^{\ell}>t]$ in terms of $\ell$. Is there any reference for this computation?