# Dependency of fundamental solution on coefficients of heat equation

Let $$b: \mathbb R_+\to\mathbb R_+$$ and $$\sigma: \mathbb R_+\times \mathbb R\to\mathbb R_+$$ be Lipschitz and bounded. Assume further $$\sigma$$ is elliptic, i.e. $$\inf_{(t,x)}\sigma(t,x)>0$$. 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

$$\partial_t g_{\alpha}(t,x,s,y) + \frac{\sigma(t,x)^2}{2\big(1+\alpha(t)\big)^2}\partial^2_{xx}g_{\alpha}(t,x,s,y) + b(t)\partial_{x}g_{\alpha}(t,x,s,y)=0,\quad \forall t\in [0,s),~ x\in\mathbb R$$

together with $$g_\alpha(s,x,s,y)=\delta_y(x)$$, where $$\alpha:\mathbb R_+\to\mathbb [0,1]$$ is Holder continuous. Does there exist a continuous function $$\theta:\mathbb R_+\to \mathbb R_+$$ with $$\theta(0)=0$$ s.t.

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

and

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

where $$\alpha,\beta: \mathbb R_+\to [0,1]$$ are both Holder continous and $$\|\alpha-\beta\|_s:=\max_{0\le u\le s}|\alpha(u)-\beta(u)|$$.

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

$$dX_t=b(t)dt + \frac{\sigma(t,X_t)}{1+\alpha(t)}dW_t,\quad \forall t\ge 0.$$

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

$$\mathbb P[X_s\in dy|X_t=x]\sim g_{\alpha}(t,x,s,y)dy.$$

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

• It looks to me like you could differentiate the equation in the variable alpha and get an equation for $\partial_\alpha g_\alpha$, with some terms that you can throw on the right side. It that looks to me like it should be relatively ok. Presumably you could get a bound in L^1 for this function, especially if you only want an estimate only up to a fixed finite time $T$. Did you try that? Commented Feb 4, 2022 at 2:40
• @ScottArmstrong Thanks a lot for the hint which seems very appealing. I didn't think of that, but the differentiation w.r.t. $\alpha$ may yield the terms $\sigma\partial_{\alpha}\sigma\partial_{xx}g_{\alpha}$ and $\partial_{\alpha}b\partial_{x}g_{\alpha}$ which contain $\partial_{x}g_{\alpha}$ and \partial_{xx}g_{\alpha}. How could we handle these terms by putting them on the right side? I suppose I don't know a priori any estimate on them... Could you please detail your idea by writting down an answer? I do appreciate! Commented Feb 4, 2022 at 7:20
• Have you tried the Malliavin calculus?
– user420828
Commented Feb 6, 2022 at 12:19
• @Philo18 Could you please specify a bit more? I know very few on Malliavin calculus Commented Feb 6, 2022 at 13:23
• @GJC20 I think you might refer to the book "The Malliavin Calculus and Related Topics", while the non constant term in front of $W$ makes your problem more complicated...
– user420828
Commented Feb 7, 2022 at 18:52