The diffusion satisfying Wentzell's boundary condition and the Markov process on the boundary, II

A semilinear equation with generalized Wentzell boundary condition

“…Since then, the finite time blow-up of solutions has been extensively investigated for the Dirichlet and Neumann boundary conditions (see [2][3][4][5][6], and the references therein). The motivation to study this equation with the generalized Wentzell boundary conditions stems from the understanding that from the probabilistic point of view, the Wentzell boundary conditions are the most general admissible boundary conditions of diffusion equations, since they include Dirichlet, Neumann, Robin and mixed boundary conditions as special cases [7][8][9][10][11]. Recently, Diaz and Tello [12][13][14] employed the Wentzell boundary conditions in modeling climatology.…”

Semigroups and boundary value problems

“…By virtue of the transversality condition (2.3), we find that a Markovian particle starting at any point of ∂ D does not stay in the boundary ∂ D all the time and enters the interior D some time or other. Probabilistically, this means that a Markov process on ∂ D is the "trace" on the boundary of trajectories of a Markov process on D (see [24]).…”

Application of Additive Functionals to the Boundary Problem of Markov Processes (Levy's System of U-Processes)

“…Therefore, a = lim ;lk and xe = lim x,, if a < oo, a.e. Therefore, we have the following: By (7), (9) PROOF. If xt_ e V and Xt G D, then by [4.10], t = T(S-) or T(S) for some s.…”

“…For given M, the U-process of M (introduced by T. Ueno) is obtained as follows (see [9] and [5]). Let M be a process satisfying (M.1), (M.2), and [2.2].…”

“…Then there exists a unique continuous additive functional cb such that (8) E e('o e d7t' ) = E-(f e-tdt) = H^Gl1(x); that is, ci = i where T is the additive functional T(t) = t A ¢, and in general, for a continuous additive functional A, such that E.(fo e-at dA) < , we define a continuous additive functional Aa by E_(fo' e-a dla) = E.(f' e-at dA) (see [4]). The functional ci satisfies (9) XV4 -f Oxv d-D_ ;I?. Now set r(s) = sup {t: ci(t) < s}; then the process 191 on V defined by (x(,J), Pt), (t E V) is called the U-process of M on V. The Green kernel Ox of 12I is given by (10) tf(t) = Et(fo' e-8f(x7()) ds) = E(fo eX-(t)f(xg) d), for every X> 0, tE V and f EB(V).…”