%0 Journal Article %1 doi:10.1137/090766152 %A Epstein, Charles L. %A Mazzeo, Rafe %D 2010 %J SIAM Journal on Mathematical Analysis %K PDE Wright-Fisher_diffusion potential_theory spectral_theory %N 2 %P 568-608 %R 10.1137/090766152 %T Wright–Fisher Diffusion in One Dimension %U http://dx.doi.org/10.1137/090766152 %V 42 %X We analyze the diffusion processes associated to equations of Wright–Fisher type in one spatial dimension. These are associated to the degenerate heat equation $\partial_tu=a(x)\partial_x^2u+b(x)\partial_xu$ on the interval $0,1$, where $a(x)>0$ on the interior and vanishes simply at the end points and $b(x)\partial_x$ is a vector field which is inward pointing at both ends. We consider various aspects of this problem, motivated by their applications in biology, including a sharp regularity theory for the “zero flux” boundary conditions, as well as an analysis of the infinitesimal generators of the $C^m$-semigroups and their adjoints. Using these results we obtain precise asymptotics of solutions of this equation, both as $t\to0,ınfty$ and as $x\to0,1$.

@article{doi:10.1137/090766152, abstract = { We analyze the diffusion processes associated to equations of Wright–Fisher type in one spatial dimension. These are associated to the degenerate heat equation $\partial_{t}u=a(x)\partial_{x}^{2}u+b(x)\partial_{x}u$ on the interval $[0,1]$, where $a(x)>0$ on the interior and vanishes simply at the end points and $b(x)\partial_{x}$ is a vector field which is inward pointing at both ends. We consider various aspects of this problem, motivated by their applications in biology, including a sharp regularity theory for the “zero flux” boundary conditions, as well as an analysis of the infinitesimal generators of the $\mathcal{C}^{m}$-semigroups and their adjoints. Using these results we obtain precise asymptotics of solutions of this equation, both as $t\to0,\infty$ and as $x\to0,1$.}, added-at = {2015-09-28T23:46:51.000+0200}, author = {Epstein, Charles L. and Mazzeo, Rafe}, biburl = {https://www.bibsonomy.org/bibtex/2ed9e52522c25900fc8e32ac4e7683967/peter.ralph}, doi = {10.1137/090766152}, eprint = {http://dx.doi.org/10.1137/090766152}, interhash = {72056f92126cc6088931635244e8ec39}, intrahash = {ed9e52522c25900fc8e32ac4e7683967}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {PDE Wright-Fisher_diffusion potential_theory spectral_theory}, number = 2, pages = {568-608}, timestamp = {2015-09-28T23:46:51.000+0200}, title = {Wright–Fisher Diffusion in One Dimension}, url = {http://dx.doi.org/10.1137/090766152}, volume = 42, year = 2010 }