Diffusions With Holding And Jumping Boundaries

For the diffusion equation u_t=u_xx in a finite interval(r₀,r₁) it is customary to add to the initial value condition u(0,x)=f(x) two boundary conditions of the form p(?)u(t,r(?))+(-1)ⁱq_iu_x(t,r(?))=0,i=0,1,and this is called the classical boundary conditions.However,various probability considerations and a long outstanding problem in genetics made it increasingly clear that,in addition to the classical type of boundary conditions there exists some of new type.Physically speaking,there exist diffusion processes where a particle can be absorbed,and stay absorbed for a finite time,after which it penetrate slowly back into the interior.Feller(1954) identified one of the new boundary conditions as the one describing this phenomenon and introduced finite sojourn time at the boundary for one dimensional diffusion processes and referred to such a process as "elementary return process".This is an important process because we can describe the totality of all stochastic process connected with Kolmolgrov backward equation by superimposing the "elementary return process to any elastic barrier process".Karlin(1981) showed that for a diffusion process,nonzero sojourn epochs could not transpire at interior points of the state space,which means that the exponential holding time can only occurs on the boundary.Based on these,we consider a process in D(?)R^d obtained in the following manner.The process coincides with the diffusion in D until it hits the boundary. Whenζ∈(?)D is reached,the process waits there for an independent random duration following an exponential distribution and leavesζby a jump into D according to the distribution v_ξand starts the diffusion afresh.The same mechanism is repeated independently each time the process reaches the boundary.The resulting diffusion will be called diffusion with holding and jumping from the boundary(DHJ).Note that due to the exponential holding time,such a process is still a Markov process,but it is irreversible.By resolvent method,we obtain the generator which can uniquely determine the DHJ process.We prove that the distribution of the process converges exponentially fast to its stationary distribution and characterize the exponent as the spectral gap for some differential operator.From PDE point of view,the DHJ process corresponds to both non-local and viscosity boundary conditions.Concretely,the organization of this thesis is as follows.Chapter 1 gives the background,current research process of relatedproblems and summarizes this thesis's work.In chapter 2,we study the Brownian motion with holding and jumping on the boundary.We use the resolvent method to obtain the infinitesimal generator because the domain of the infinitesimal generator is essentially the same as the range of the resolvent.Knowledge of this range and of the differential operator determines uniquely the infinitesimal generator.Since the semigroup generated by the DHJ is not strongly continuous,to use the nice property of strongly continuous semigroup in analytic theory,in chapter 3 we show that the dual is strongly continuous and derive ergodicity through spectral radius formulas and finally obtain the ergodic theorem by duality. In chapter 4,we discuss a class of a more general process---one dimensional Feller diffusion proposed by W.Feller in 1954.The Feller diffusion allows the possibility of jumps from boundary to boundary,not only from boundary to the interior.We give the stationary distribution of this process.In Chapter 5,some related problems are discussed.The PDE aspect of DHJ and the relationship of eigenvalues with convergence speed is investigated.Finally,we discuss the first passage problem for one dimensional Feller diffusion and obtain the Laplace-Stieltjes transformation of the first passage time.