Suppose that X is a right process which is associated with a semi-Dirichlet form （ε, D（ε）） on L2（E; m）. Let J be the jumping measure of （ε, D（ε）） satisfying J（E x E- d） 〈 ∞. Let u E D（ε）b ：= D（...Suppose that X is a right process which is associated with a semi-Dirichlet form （ε, D（ε）） on L2（E; m）. Let J be the jumping measure of （ε, D（ε）） satisfying J（E x E- d） 〈 ∞. Let u E D（ε）b ：= D（ε） N L（E; m）, we have the following Pukushima＇s decomposition u（Xt）-u（X0） --- Mut ＋ Nut. Define Pu f（x） = Ex[eNT f（Xt）]. Let Qu（f,g） = ε（f,g）＋ε（u, fg） for f, g E D（ε）b. In the first part, under some assumptions we show that （Qu, D（ε）b） is lower semi-bounded if and only if there exists a constant a0 〉 0 such that /Put/2 ≤eaot for every t 〉 0. If one of these assertions holds, then （Put〉0is strongly continuous on L2（E;m）. If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J（E x E - d） 〈 ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of At and give two sufficient conditions for PAf（x） = Ex[eAtf（Xt）] to be strongly continuous.展开更多
Anomalous diffusions are ubiquitous in nature,whose functional distributions are governed by the backward Feynman-Kac equation.In this paper,the local discontinuous Galerkin(LDG)method is used to solve the 2D backward...Anomalous diffusions are ubiquitous in nature,whose functional distributions are governed by the backward Feynman-Kac equation.In this paper,the local discontinuous Galerkin(LDG)method is used to solve the 2D backward Feynman-Kac equation in a rectangular domain.The spatial semi-discrete LDG scheme of the equivalent form(obtained by Laplace transform)of the original equation is established.After discussing the properties of the fractional substantial calculus,the stability and optimal convergence rates of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux.The L1 scheme on the graded meshes is used to deal with the weak singularity of the solution near the initial time.Based on the theoretical results of a semi-discrete scheme,we investigate the stability and convergence of the fully discrete scheme,which shows the optimal convergence rates.Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme.In addition,we also verify the effect of the central numerical flux on the convergence rates and the condition number of the coefficient matrix.展开更多
We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. Thi...We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.展开更多
Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We s...Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup展开更多
Parabolic equations on general bounded domains are studied. Usingthe refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE's ...Parabolic equations on general bounded domains are studied. Usingthe refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE's method has the Feynmann_Kac representation for any bounded domains.展开更多
LetXbe an m s ymmetric Markov process andMa multiplicative functional ofXsuch that theMsubprocess ofXis alsom-symmetric.The author characterizes the Dirichlet form associated with the subprocess in terms of that assoc...LetXbe an m s ymmetric Markov process andMa multiplicative functional ofXsuch that theMsubprocess ofXis alsom-symmetric.The author characterizes the Dirichlet form associated with the subprocess in terms of that associated withXand the bivariate Revuz measure ofM.展开更多
基金supported by NSFC(11201102,11326169,11361021)Natural Science Foundation of Hainan Province(112002,113007)
文摘Suppose that X is a right process which is associated with a semi-Dirichlet form （ε, D（ε）） on L2（E; m）. Let J be the jumping measure of （ε, D（ε）） satisfying J（E x E- d） 〈 ∞. Let u E D（ε）b ：= D（ε） N L（E; m）, we have the following Pukushima＇s decomposition u（Xt）-u（X0） --- Mut ＋ Nut. Define Pu f（x） = Ex[eNT f（Xt）]. Let Qu（f,g） = ε（f,g）＋ε（u, fg） for f, g E D（ε）b. In the first part, under some assumptions we show that （Qu, D（ε）b） is lower semi-bounded if and only if there exists a constant a0 〉 0 such that /Put/2 ≤eaot for every t 〉 0. If one of these assertions holds, then （Put〉0is strongly continuous on L2（E;m）. If X is equipped with a differential structure, then under some other assumptions, these conclusions remain valid without assuming J（E x E - d） 〈 ∞. Some examples are also given in this part. Let At be a local continuous additive functional with zero quadratic variation. In the second part, we get the representation of At and give two sufficient conditions for PAf（x） = Ex[eAtf（Xt）] to be strongly continuous.
基金supported by the National Natural Science Foundation of China under Grant No.12225107the Major Science and Technology Projects in Gansu Province Leading Talents in Science and Technology under Grant No.23ZDKA0005+1 种基金the Innovative Groups of Basic Research in Gansu Province under Grant No.22JR5RA391Lanzhou Talent Work Special Fund.
文摘Anomalous diffusions are ubiquitous in nature,whose functional distributions are governed by the backward Feynman-Kac equation.In this paper,the local discontinuous Galerkin(LDG)method is used to solve the 2D backward Feynman-Kac equation in a rectangular domain.The spatial semi-discrete LDG scheme of the equivalent form(obtained by Laplace transform)of the original equation is established.After discussing the properties of the fractional substantial calculus,the stability and optimal convergence rates of the semi-discrete scheme are proved by choosing an appropriate generalized numerical flux.The L1 scheme on the graded meshes is used to deal with the weak singularity of the solution near the initial time.Based on the theoretical results of a semi-discrete scheme,we investigate the stability and convergence of the fully discrete scheme,which shows the optimal convergence rates.Numerical experiments are carried out to show the efficiency and accuracy of the proposed scheme.In addition,we also verify the effect of the central numerical flux on the convergence rates and the condition number of the coefficient matrix.
基金supported by National Natural Science Foundation of China(Grant No.10921101)the Programme of Introducing Talents of Discipline to Universities of China(Grant No.B12023)the Fundamental Research Funds of Shandong University
文摘We introduce a new type of path-dependent quasi-linear parabolic PDEs in which the continuous paths on an interval [0, t] become the basic variables in the place of classical variables （t, x） ∈[0, T]× R^d. This new type of PDEs are formulated through a classical BSDE in which the terminal values and the generators are allowed to be general function of Brownian motion paths. In this way, we establish the nonlinear Feynman- Kac formula for a general non-Markoviau BSDE. Some main properties of solutions of this new PDEs are also obtained.
基金The authors would like to thank Professor Mu-Fa Chen and Professor Feng-Yu Wang for introducing them the field of functional inequalities when they studied in Beijing Normal University, and for their continuous encouragement and great help in the past few years. The authors are also indebted to the referees for valuable comments on the draft. This work was supported by the National Natural Science Foundation of China （Grant No. 11201073）, Japan Society for the Promotion of Science （No. 26.04021）, the Natural Science Foundation of Fujian Province （No. 2015J01003）, and the Program for Nonlinear Analysis and Its Applications （No. IRTL1206） （for Jian Wang）.
文摘Let （Xt）t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form （D, （D）） as follows：where J（x, y） is a strictly positive and symmetric measurable function on Rd × Rd. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup
基金This work was supported by the National Natural Science Foundation of China ( Grant No. 19631020) Mathematical Center of the Ministry of Education of China, Part of the work was completed while the authors were visiting Department of Mathematics, the
文摘Parabolic equations on general bounded domains are studied. Usingthe refined maximum principle, existence and the semigroup property of solutions are obtained. It is also shown that the solution obtained by PDE's method has the Feynmann_Kac representation for any bounded domains.
文摘LetXbe an m s ymmetric Markov process andMa multiplicative functional ofXsuch that theMsubprocess ofXis alsom-symmetric.The author characterizes the Dirichlet form associated with the subprocess in terms of that associated withXand the bivariate Revuz measure ofM.
