The following questiion comes from the article of Stroock & Varadhan (Diffusion processes with continuous coefficients I - 1969 - pg 378 )

We consider the operator $G_\lambda$ $$G_\lambda f(s,x) = \int_s^\infty \int_{\mathbb{R}^d} \frac{1}{\big(2\pi (t-s)\big)^{d/2}} \exp\big\{-\frac{\vert y-x \vert^2}{2(t-s)} \big\}e^{-\lambda(t-s)}f(t,y)\, dy \,dt$$


$$g_\lambda(s,x) = \frac{1}{\big(2\pi (s)\big)^{d/2}}\exp\big\{-\frac{\vert x \vert^2}{2(s)} \big\}e^{-\lambda(t-s)} 1_{s>0}$$

therefore we can view $G_\lambda f(s,x)$ as a convolution (in $\mathbb{R}\times \mathbb{R}^d$) $$g_\lambda * f (s,x) = \int_0^\infty \int_{\mathbb{R}^d} g_\lambda(t-s,y-x) f(t,y) \,dy \, dx$$

One can check that $\vert\vert g_\lambda \vert\vert_{L^q} < \infty$ for q < (d+2)/d (it suffices to integrate in space, one obtains $C_{\lambda,q}\int_0^\infty t^{\phi(q)} e^{-t}\,dt$, now just check when $\phi(\alpha) = - \frac{d}{2} (\alpha-1)> -1$ this yields the result)

Similarly one can see that $\vert\vert \nabla g_\lambda \vert\vert_{L^q} < \infty$ for $q < (d+2)/(d+1)$

Using Hölder's inequality one can see that for $ p : \frac{1}{p} + \frac{1}{q} = 1 $ and $q<(d+2)/d$ $$\sup_{s,x} \big\vert(G_\lambda) f(s,x)\big\vert \leq C_{\lambda, p} \vert\vert f\vert\vert_p$$

Moreover for q < (d+2)/(d+1) Hölder inequality gives us that ($\frac{1}{p} +\frac{1}{q} = 1$) $$ \big\vert G_\lambda f (s,x_1) - G_\lambda(s,x_2) \big\vert \leq \bar{C}_{\lambda, p} \vert\vert f\vert\vert_p \vert x_1-x_2\vert$$

So far I have followed,

Now the authors make two claims:

"Moreover, $G_\lambda f $ has uniformly continuous first derivatives in $x$


\begin{multline} \big \vert G_\lambda f (s,x_1 + h) - G_\lambda(s,x_1) - G_\lambda f (s,x_2 + h) + G_\lambda(s,x_2)\big \vert \leq w_0(\vert h\vert) \bar{C}_{\lambda,p}\vert\vert f\vert\vert_p \vert x_1-x_2\vert \end{multline} where $w_0(\vert h\vert) \downarrow 0 $ as $\vert h\vert \downarrow 0$ "

to see

$G_\lambda f $ has uniformly continuous first derivatives

I tried $$ \big \vert \partial_i G_\lambda f (s,x) - \partial_i G_\lambda(s',x') \big \vert = \big \vert \lim_{h \to 0} h^{-1}[G_\lambda f (s,x + h e_i ) - G_\lambda f (s',x' + h e_i ) - (G_\lambda f (s,x) - G_\lambda f (s',x' )] \big \vert \\ =\big \vert \lim_{h \to 0} h^{-1}[g_\lambda *(f - \tau_v f) (s,x + h e_i ) - g_\lambda *(f - \tau_v f)_\lambda(s,x) ] \big \vert \\ \leq \bar{C}_{\lambda,p} \vert \vert f - \tau_v f \vert \vert_p \underset{v \to 0}{\longrightarrow} 0 $$

where $v = (s' -s , x' -x)$ and $\tau_v f (a,b) = f(a + s'-s,b + x'- x)$

To the last claim however I couldn't arrive at any answer, unless

$$\vert \vert f - \tau_v f\vert \vert_p \leq w_{0}(\vert h\vert )\,\vert \vert f\vert \vert_p $$

But I think this is not true.

Any ideas?


We can find that \begin{align*} &\big \vert G_\lambda f (s,x_1 + h) - G_\lambda(s,x_1) - G_\lambda f (s,x_2 + h) + G_\lambda(s,x_2)\big \vert = \vert G_\lambda'(s,\tilde{x}_1)h - G_\lambda'(s,\tilde{x}_2)h\big \vert\\ &\leq h C_{\lambda,q}\|g'_\lambda\|_q \|f - \tau_{\tilde{x}_2 - \tilde{x}_1}f\|_p \end{align*}

And consider the following as a bad typo:enter image description here


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.