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. $\inf_{(t,x,\alpha)}\sigma(t,x,\alpha)>0$. For each $s>0$ and $y\in\mathbb R$, let $g_{\alpha}(\cdot,\cdot,s,y):[0,s]\times \mathbb R\to\mathbb R$ be the fundamental solution to

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

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

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

and

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

Any answer, comments and references are highly appreciated.

PS : A probabilistic interprectation of $g_{\alpha}$ is as follows : Consider the parametric stochastic differential equation (SDE)

$$dX_t=b(t,X_t)dt + \sigma(t,X_t,\alpha)dW_t,\quad \forall t\ge 0.$$

Then $g_{\alpha}$ is the conditional density of $X_s$ knowing that $X_t=x$, i.e.

$$\mathcal L(X_s|X_t=x)\sim g_{\alpha}(t,x,s,y)dy.$$

However, I don't know any probability tools to handle the density regularity of the solution to a SDE...