Quasistationarity and extinction for population processes under asymptotic reversibility conditions

We consider stochastic population processes that are almost surely absorbed at the origin within finite time. Our interest is in the quasistationary distribution, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi></mml:mrow> </mml:math> , and the expected time, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math> , from quasistationarity to extinction, both of which we study via WKB approximation. This approach involves solving a Hamilton-Jacobi partial differential equation specific to the model. We provide conditions under which analytical solution of the Hamilton-Jacobi equation is possible, and give the solution. This provides a first approximation to both <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi></mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math> . We provide further conditions under which a corresponding 'transport equation' may be solved, leading to an improved approximation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi></mml:mrow> </mml:math> . For multitype birth and death processes, we then consider an alternative approximation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>u</mml:mi></mml:mrow> </mml:math> that is valid close to the origin, provide conditions under which the elements of this alternative approximation may be found explicitly, and hence derive an improved approximation for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>τ</mml:mi></mml:math> . We illustrate our results in a number of applications.