With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the stead...With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation.In this paper,time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process.Based on the original steady-state diffusion equation,we first propose a stochastic partial differential equationmodel.The numerical solution to the stochastic partial differential model is carried out by using the finite element method.When the time resolution is high enough,the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation,which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality.In addition,the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide inmuscle tissue are also first proposed by GEANT4Monte Carlomethod.The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations,which shows that the stochastic partial differential equation can simulate the corresponding process.展开更多
This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information avail...This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.展开更多
We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with j...We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy's measure is infinite.展开更多
This article studies the asymptotic behaviors of the solution for a stochastic hydrodynamical equation in Heisenberg paramagnet in a two-dimensional periodic domain. We obtain the existence of random attractors in H1.
This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covarianc...This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.展开更多
In this paper we study a fractional stochastic heat equation on Rd (d 〉 1) with additive noise /t u(t, x) = Dα/δ u(t, x)+ b(u(t, x) ) + WH (t, x) where D α/δ is a nonlocal fractional differential...In this paper we study a fractional stochastic heat equation on Rd (d 〉 1) with additive noise /t u(t, x) = Dα/δ u(t, x)+ b(u(t, x) ) + WH (t, x) where D α/δ is a nonlocal fractional differential operator and W H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d = 1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.展开更多
This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the stand...This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.展开更多
Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical method...Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.展开更多
In this paper, a class of nonlinear stochastic partial differential equations with discontinuous coefficients is investigated. This study is motivated by some research on stochastic viscosity solutions under non-Lipsc...In this paper, a class of nonlinear stochastic partial differential equations with discontinuous coefficients is investigated. This study is motivated by some research on stochastic viscosity solutions under non-Lipschitz conditions recently. By studying the solutions of backward doubly stochastic differential equations with discontinuous coefficients and constructing a new approximation function <em>f</em><sub><em>n</em></sub> to the coefficient <em>f</em>, we get the existence of stochastic viscosity sub-solutions (or super-solutions).The results of this paper can be seen as the extension and application of related articles.展开更多
In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of referen...In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.展开更多
In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness o...In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness of these solutions were established.The penalization method plays an important role.We also provided a probabilistic interpretation of the classical solutions of the mean-reflected stochastic partial differential equations(MR-SPDEs)in terms of MR-BDSDEs.展开更多
In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.
We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition....We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.展开更多
A conceptual model for microscopic-macroscopic slow-fast stochastic systems is considered. A dynamical reduction procedure is presented in order to extract effective dynamics for this kind of systems. Under appropriat...A conceptual model for microscopic-macroscopic slow-fast stochastic systems is considered. A dynamical reduction procedure is presented in order to extract effective dynamics for this kind of systems. Under appropriate assumptions, the effective system is shown to approximate the original system, in the sense of a probabilistic convergence.展开更多
In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions,by the way,we obtain the quasi-sure continuity of the solut...In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions,by the way,we obtain the quasi-sure continuity of the solution.展开更多
This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stoc...This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.展开更多
In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to se...In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.展开更多
We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws.The equation is driven by double-para...We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws.The equation is driven by double-parameter fractional noises.In addition,the existence and moment estimate are also obtained for the density of the law of such a solution.展开更多
基金National Science Foundation of China(NSFC)(61671009,12171178).
文摘With the development of molecular imaging,Cherenkov optical imaging technology has been widely concerned.Most studies regard the partial boundary flux as a stochastic variable and reconstruct images based on the steadystate diffusion equation.In this paper,time-variable will be considered and the Cherenkov radiation emission process will be regarded as a stochastic process.Based on the original steady-state diffusion equation,we first propose a stochastic partial differential equationmodel.The numerical solution to the stochastic partial differential model is carried out by using the finite element method.When the time resolution is high enough,the numerical solution of the stochastic diffusion equation is better than the numerical solution of the steady-state diffusion equation,which may provide a new way to alleviate the problem of Cherenkov luminescent imaging quality.In addition,the process of generating Cerenkov and penetrating in vitro imaging of 18 F radionuclide inmuscle tissue are also first proposed by GEANT4Monte Carlomethod.The result of the GEANT4 simulation is compared with the numerical solution of the corresponding stochastic partial differential equations,which shows that the stochastic partial differential equation can simulate the corresponding process.
文摘This paper considers a mean-field type stochastic control problem where the dynamics is governed by a forward and backward stochastic differential equation (SDE) driven by Lévy processes and the information available to the controller is possibly less than the overall information. All the system coefficients and the objective performance functional are allowed to be random, possibly non-Markovian. Malliavin calculus is employed to derive a maximum principle for the optimal control of such a system where the adjoint process is explicitly expressed.
文摘We consider the problem of viscosity solution of integro-partial differential equation( IPDE in short) with one obstacle via the solution of reflected backward stochastic dif ferential equations(RBSDE in short) with jumps. We show the existence and uniqueness of a continuous viscosity solution of equation with non local terms, if the generator is not monotonous and Levy's measure is infinite.
文摘This article studies the asymptotic behaviors of the solution for a stochastic hydrodynamical equation in Heisenberg paramagnet in a two-dimensional periodic domain. We obtain the existence of random attractors in H1.
基金supported by an NSERC granta startup fund of University of Alberta
文摘This article attempts to give a short survey of recent progress on a class of elementary stochastic partial differential equations (for example, stochastic heat equations) driven by Gaussian noise of various covariance structures. The focus is on the existence and uniqueness of the classical (square integrable) solution (mild solution, weak solution). It is also concerned with the Feynman-Kac formula for the solution;Feynman-Kac formula for the moments of the solution;and their applications to the asymptotic moment bounds of the solution. It also briefly touches the exact asymptotics of the moments of the solution.
基金Supported by NNSFC(11401313)NSFJS(BK20161579)+2 种基金CPSF(2014M560368,2015T80475)2014 Qing Lan ProjectSupported by MEC Project PAI80160047,Conicyt,Chile
文摘In this paper we study a fractional stochastic heat equation on Rd (d 〉 1) with additive noise /t u(t, x) = Dα/δ u(t, x)+ b(u(t, x) ) + WH (t, x) where D α/δ is a nonlocal fractional differential operator and W H is a Gaussian-colored noise. We show the existence and the uniqueness of the mild solution for this equation. In addition, in the case of space dimension d = 1, we prove the existence of the density for this solution and we establish lower and upper Gaussian bounds for the density by Malliavin calculus.
基金partially supported by the National Science and Engineering Research Council of Canada(NSERC)the start-up funds from the University of Calgary
文摘This article is devoted to the study of fully nonlinear stochastic Hamilton-Jacobi (HJ) equations for the optimal stochastic control problem of ordinary differential equations with random coefficients. Under the standard Lipschitz continuity assumptions on the coefficients, the value function is proved to be the unique viscosity solution of the associated stochastic HJ equation.
文摘Stochastic multi-symplectic methods are a class of numerical methods preserving the discrete stochastic multi-symplectic conservation law. These methods have the remarkable superiority to conventional numerical methods when applied to stochastic Hamiltonian partial differential equations (PDEs), such as long-time behavior, geometric structure preserving, and physical properties preserving. Stochastic Maxwell equations driven by either additive noise or multiplicative noise are a system of stochastic Hamiltonian PDEs intrinsically, which play an important role in fields such as stochastic electromagnetism and statistical radiophysics. Thereby, the construction and the analysis of various numerical methods for stochastic Maxwell equations which inherit the stochastic multi-symplecticity, the evolution laws of energy and divergence of the original system are an important and promising subject. The first stochastic multi-symplectic method is designed and analyzed to stochastic Maxwell equations by Hong et al.(A stochastic multi-symplectic scheme for stochastic Maxwell equations with additive noise. J. Comput. Phys. 268:255-268, 2014). Subsequently, there have been developed various stochastic multi-symplectic methods to solve stochastic Maxwell equations. In this paper, we make a review on these stochastic multi-symplectic methods for solving stochastic Maxwell equations driven by a stochastic process. Meanwhile, the theoretical results of well-posedness and conservation laws of the stochastic Maxwell equations are included.
文摘In this paper, a class of nonlinear stochastic partial differential equations with discontinuous coefficients is investigated. This study is motivated by some research on stochastic viscosity solutions under non-Lipschitz conditions recently. By studying the solutions of backward doubly stochastic differential equations with discontinuous coefficients and constructing a new approximation function <em>f</em><sub><em>n</em></sub> to the coefficient <em>f</em>, we get the existence of stochastic viscosity sub-solutions (or super-solutions).The results of this paper can be seen as the extension and application of related articles.
基金supported by the National Natural Science Foundations of China(Grant Nos.12071261,11831010)the National Key R&D Program(Grant No.2018YFA0703900).
文摘In this paper,we explore a new approach to design and analyze numerical schemes for backward stochastic differential equations(BSDEs).By the nonlinear Feynman-Kac formula,we reformulate the BSDE into a pair of reference ordinary differential equations(ODEs),which can be directly discretized by many standard ODE solvers,yielding the corresponding numerical schemes for BSDEs.In particular,by applying strong stability preserving(SSP)time discretizations to the reference ODEs,we can propose new SSP multistep schemes for BSDEs.Theoretical analyses are rigorously performed to prove the consistency,stability and convergency of the proposed SSP multistep schemes.Numerical experiments are further carried out to verify our theoretical results and the capacity of the proposed SSP multistep schemes for solving complex associated problems.
文摘In this paper,we study a class of mean-reflected backward doubly stochastic differential equations(MR-BDSDEs),where the constraint depends on the law of the solution and not on its paths.The existence and uniqueness of these solutions were established.The penalization method plays an important role.We also provided a probabilistic interpretation of the classical solutions of the mean-reflected stochastic partial differential equations(MR-SPDEs)in terms of MR-BDSDEs.
基金Supported by the LPMC at Nankai Universitythe NSF of China (Grant No. 10871103)
文摘In this paper, we prove a large deviation principle for a class of stochastic Cahn-Hilliard partial differential equations driven by space-time white noises.
基金This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11571190,11771218,11771018,12061004)the Natural Science Foundation of Ningxia(No.2020AAC03230)the Major Research Project for North Minzu University(No.ZDZX201902).
文摘We establish a class of stochastic partial differential equations (SPDEs) driven by space-time fractional noises, where we suppose that the drfit term contains a gradient and satisfies certain non-Lipschitz condition. We prove the strong existence and uniqueness and joint Hölder continuity of the solution to the SPDEs.
基金supported by NSF of China (10901065, 10971225, and11028102)the NSF Grants 1025422 and 0731201the Cheung Kong Scholars Program, and an open research grant from the State Key Laboratory for Nonlinear Mechanics at the Chinese Academy of Sciences
文摘A conceptual model for microscopic-macroscopic slow-fast stochastic systems is considered. A dynamical reduction procedure is presented in order to extract effective dynamics for this kind of systems. Under appropriate assumptions, the effective system is shown to approximate the original system, in the sense of a probabilistic convergence.
基金This work is supported by NSF(No.10301011)of China and Project 973
文摘In this paper we prove a quasi-sure limit theorem of parabolic stochastic partial differential equations with smooth coefficients and some initial conditions,by the way,we obtain the quasi-sure continuity of the solution.
基金Project supported by the National Natural Science Foundation of China (No.10325101, No.101310310)the Science Foundation of the Ministry of Education of China (No. 20030246004).
文摘This paper explores the diffeomorphism of a backward stochastic ordinary differential equation (BSDE) to a system of semi-linear backward stochastic partial differential equations (BSPDEs), under the inverse of a stochastic flow generated by an ordinary stochastic differential equation (SDE). The author develops a new approach to BSPDEs and also provides some new results. The adapted solution of BSPDEs in terms of those of SDEs and BSDEs is constructed. This brings a new insight on BSPDEs, and leads to a probabilistic approach. As a consequence, the existence, uniqueness, and regularity results are obtained for the (classical, Sobolev, and distributional) solution of BSPDEs.The dimension of the space variable x is allowed to be arbitrary n, and BSPDEs are allowed to be nonlinear in both unknown variables, which implies that the BSPDEs may be nonlinear in the gradient. Due to the limitation of space, however, this paper concerns only classical solution of BSPDEs under some more restricted assumptions.
基金supported by National Natural Science Foundation of China (Grant Nos. 10971076, 10871215)
文摘In this paper, we study the regularity of solutions of nonlinear stochastic partial differential equations (SPDEs) with multiplicative noises in the framework of Hilbert scales. Then we apply our abstract result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau equations on the real line, stochastic 2D Navier-Stokes equations (SNSEs) in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their smooth solutions respectively. In particular, we also get the existence of local smooth solutions for 3D SNSEs.
基金supported by Mathematical Tianyuan Foundation of China(Grant No.11226198)Priority Academic Program Development of Jiangsu Higher Education Institutions+1 种基金National Natural Science Foundation of China(Grant No.11171062)Innovation Program of Shanghai Municipal Education Commission(Grant No.12ZZ063)
文摘We study the existence,uniqueness and Hlder regularity of the solution to a stochastic semilinear equation arising from 1-dimensional integro-differential scalar conservation laws.The equation is driven by double-parameter fractional noises.In addition,the existence and moment estimate are also obtained for the density of the law of such a solution.