I have ran into the following integral equation as part of my phd research project, trying to enforce a boundary condition of a parabolic pde problem.

For $\xi = (\alpha\theta)^{1/\alpha}$ and for all $\theta>0$.

I have the following equality $$\int\limits_0^\infty g(\kappa, x_0) \exp{\left[-\frac{\left(\xi + \kappa \right)^2}{2\alpha\theta}\right]} d\kappa % = \exp{\left[-\frac{\left(\xi - x_0\right)^2}{2\alpha\theta}\right]}%. $$ If we assume $g(\kappa, x_0)$ to be a smooth function dependent on $\kappa$ and $x_0$. Is there a way to find $g(\kappa, x_0)$ ? I do not have a broad enough background in integral equations, so will really appreciate some guidance.

I was also wondering if the following technique can be applied here Power series solution for integral equations wikipedia

Thank you.

PS: I also posted on Math Stack Exchange Post but my post did not generate much interest from others.

Edit 1: This condition can also be presented as follows $$ \int\limits_0^\infty g(\kappa)\exp{\left(-\kappa^2/(2\alpha \theta) -\xi\kappa/(\alpha\theta)\right)} d\kappa = \exp{\left(-\frac{x_0^2 - 2\xi x_0}{2\alpha\theta}\right)}$$ Can it be interpreted as some form of convolution ?