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This post can be thought of as the prototype proof and the motivation for the question posted here Using a laplace type expansion to get bounds on an integral arising in the study of Brownian motion

Let $ 0 < r < 1$, fix $x > 1$ and consider the integral

$$ I_{1}(x) = \int_{1}^{\infty} \exp\left( - \frac{x^2}{2y^{2r}} - \frac{y^2}{2}\right) \frac{dy}{y^r}.$$

The result we have been trying to prove below can be thought of as an analysis of localizing around the maximum of the function $f(y) = \frac{x^2}{2y^{2r}} + \frac{y^2}{2}$ in the exponential of the integrand and then applying appropriate estimates along with integration by parts. Roughly speaking I am having trouble with the analysis on one side of the maximum.

Let $I_1(x) = \int_{1}^{\infty} \exp( -\frac{x^2}{2y^{2r}} - \frac{y^2}{2} ) \frac{dy}{y^r}$ and fix a constant $c^* = r^{\frac{1}{2r+2}} $ that is $c^* = argmin ( d(c) )$ where $d(c) = \frac{1}{2} (c^{-2r} + c^2)$ and let $x^* = x^{\frac{1}{1+r}}$. Then we break up the integral as

$$I_1(x): = I_{11}(x) + I_{12}(x) = \int_{1}^{c^* x^*} \exp( -\frac{x^2}{2y^{2r}} - \frac{y^2}{2} ) \frac{dy}{y^r} + \int_{c^*x^*}^{\infty} \exp( -\frac{x^2}{2y^{2r}} - \frac{y^2}{2} ) \frac{dy}{y^r} $$

The particular trouble I am having arises with estimating $I_{11}$.

Using integration by parts on $I_{11}(x)$ we set $f(y) = \frac{x^2}{2y^{2r}} + \frac{y^2}{2}$ and write $$ I_{11}(x) = \int_{1}^{c^* x^*} \exp( -\frac{x^2}{2y^{2r}} - \frac{y^2}{2} ) \frac{dy}{y^r} = \int_{1}^{c^* x^*} \frac{y^{-r}}{f'(y)} f'(y) e^{-f(y)} dy$$

Let $v = \frac{1}{y^r f'(y)}, du = f'(y) e^{-f(y)}dy$ so that $u = - e^{-f(y)}$ and $dv = \frac{-r y^{-r-1}f'(y) -y^{r} f''(y) }{(f'(y))^2}$ and write $$dv = \frac{-r y^{-r-1}}{f'(y)} dy - \frac{y^{-r} f''(y)}{(f'(y))^2}dy := v_1(y)dy + v_2(y)dy$$

Applying integration by parts to $I_{11}(x)$ we get

$$ \int_{1}^{c^* x^*} \frac{y^{-r}}{f'(y)} f'(y) e^{-f(y)} dy = \left[- \frac{y^{-r}}{f'(y)} exp(-f(y)) \right]_{1}^{c^* x^*} + \int_{1}^{c^* x^*} e^{-f(y)} (v_1(y) +v_2(y)) dy $$

1) Using this argument is it possible to show $I_{11}(x) \leq k(r,x)\exp(- x^{\frac{2}{1+r}} d(c^*))$ where $k(r,x)$ is a rational function in $x$?

The issue seems to be for us finding a way to remove the singularities around $v_1$ and $v_2$ near $c^* x^*$.

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