Let $b: \mathbb R_+\times \mathbb R_+\to\mathbb R_+$ and $\sigma: \mathbb R_+\times \mathbb R\times\mathbb R_+\to\mathbb R_+$ be Lipschitz and bounded. Assume further $\sigma$ is elliptic, i.e. $\inf_{(t,x,m)}\sigma(t,x,m)>0$. For each $s>0$ and $y\in\mathbb R$, let $g_m(\cdot,\cdot,s,y):[0,s]\times \mathbb R\to\mathbb R$ be the fundamental solution to

$$\partial_t g_m(t,x,s,y) + \frac{\sigma(t,x,m)^2}{2}\partial^2_{xx}g_m(t,x,s,y) + b(t,m)\partial_{x}g_m(t,x,s,y)=0,\quad \forall t\in [0,s),~ x\in\mathbb R$$

together with $g(s,x,s,y)=\delta_y(x)$. Fix $T>0$. Does there exist a continuous function $\theta:\mathbb R_+\to \mathbb R_+$ with $\theta(0)=0$ s.t.

$$\left|\int_0^{\infty}g_m(0,x,s,y)dy-\int_0^{\infty}g_n(0,x,s,y)dy\right|\le \theta(|m-n|),\quad \forall s\le T,~ \forall x\in\mathbb R$$

and

$$\int_0^s\left|\frac{d}{dt}\left(\int_0^{\infty}g_m(t,0,s,y)dy-\int_0^{\infty}g_n(t,0,s,y)dy\right)\right|ds\le \theta(|m-n|),\quad \forall s\le T?$$

Any answer, comments and references are highly appreciated.