I have ran into the following integral equation as part of my research.
For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$.
I have the following equality $$\int\limits_0^\infty g(\kappa) \exp{\left[-\frac{\left(\xi + \kappa \right)^2}{2\alpha\theta}\right]} d\kappa % = \exp{\left[-\frac{\left(\xi - x_0\right)^2}{2\alpha\theta}\right]}%. $$ This condition can also be presented as follows $$ \int\limits_0^\infty g(\kappa)\exp{\left(-\kappa^2/(2\alpha \theta) -\xi\kappa/(\alpha\theta)\right)} d\kappa = \exp{\left(-\frac{x_0^2 - 2\xi x_0}{2\alpha\theta}\right)}$$ I assume $\kappa$ to depend on $\theta$, Is there a way to find $g(\kappa)$ here ? Can it be interpreted as some form of convolution ? Thank you.