Yamamuro in [1] defines strong and weak transience of Markov processes; gives a criterion for strong transience of Feller processes; and further, discusses strong and weak transience of Ornstein-Uhlenbeck type process...Yamamuro in [1] defines strong and weak transience of Markov processes; gives a criterion for strong transience of Feller processes; and further, discusses strong and weak transience of Ornstein-Uhlenbeck type processes. In this article, the authors weaken the Feller property of the result in [1] to weak Feller property and discuss the strong transience of operator-self-similar Markov processes.展开更多
We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimizatio...We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimization of the ergodic(or long-term average)total coat.Eooentially,this paper trento the cnce where,for n_(t)=n fixed,z_(t)ia a reflected diffusion or a reflected diffusion with jumps,nt being,for fixed z,a continuous-time Markov chain.Using the vanishing discount appronch,we exctend existing reoulta dealing with the situation where nt evolvea only by the switching control action and the diffusion is non-degenerate.Moreover,we solve the ergodic problem for a claso of diffusiono which can be degenerate and for an example with aboorbing atate.展开更多
Let d ≥ 1 and Z be a subordinate Brownian motion on R^d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish th...Let d ≥ 1 and Z be a subordinate Brownian motion on R^d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L^b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p^b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L^b, C_c~∞(R^d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.展开更多
基金Research supported in part by the National Natural Science Foundation of China and a grant from the Ministry of Education of China
文摘Yamamuro in [1] defines strong and weak transience of Markov processes; gives a criterion for strong transience of Feller processes; and further, discusses strong and weak transience of Ornstein-Uhlenbeck type processes. In this article, the authors weaken the Feller property of the result in [1] to weak Feller property and discuss the strong transience of operator-self-similar Markov processes.
文摘We consider the control of the diserete component n_(t)of a owitching Markov proceaa x_(t)=(z_(t),n_(t))when there ia a running cost and an immediate coat c(i,j)for owitching n_(t)from i to j.We satudy the minimization of the ergodic(or long-term average)total coat.Eooentially,this paper trento the cnce where,for n_(t)=n fixed,z_(t)ia a reflected diffusion or a reflected diffusion with jumps,nt being,for fixed z,a continuous-time Markov chain.Using the vanishing discount appronch,we exctend existing reoulta dealing with the situation where nt evolvea only by the switching control action and the diffusion is non-degenerate.Moreover,we solve the ergodic problem for a claso of diffusiono which can be degenerate and for an example with aboorbing atate.
基金supported by National Science Foundation of USA(Grant No.DMS-1206276)National Natural Science Foundation of China(Grant No.11371217)
文摘Let d ≥ 1 and Z be a subordinate Brownian motion on R^d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L^b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p^b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L^b, C_c~∞(R^d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.
