Solution of one boundary value problem on the semi-axis for the diffusion equation with a growing coefficient

In [1] a similar problem has been solved by probabilistic methods. In my talk, I will speak about an approach to solving this problem based on the non-oscillating WKB method. The difficulty arising in this case is that the representation of the Dirac delta-function in the form of a non-oscillating WKB function is little (let's say) known. However, such a representation exists, and its use leads to the construction of a fundamental solution to the boundary value problem, and, in particular, to the results [1] (and more general ones). [1] V. Bansaye, P. Collet, S. Martinez, S. Méléard, J. S. Martin. Diffusions from infinity // Trans. Amer. Math. Soc., Vol. 372 (2019), pp. 5781-5823