Gelugpa is the most influential extant religious sect of Tibetan Buddhism,which is the spiritual prop for Tibetans,with thousands of monasteries and followers in Tibetan areas of China.Studies on the spatial diffusion...Gelugpa is the most influential extant religious sect of Tibetan Buddhism,which is the spiritual prop for Tibetans,with thousands of monasteries and followers in Tibetan areas of China.Studies on the spatial diffusion processes of Gelugpa can not only reveal its historical geographical development but also lay the foundation for anticipating its future development trend.However,existing studies on Gelugpa lack geographical perspective,making it difficult to explore the spatial characteristics.Furthermore,the prevailing macro-perspective overlooks spatiotemporal heterogeneity in diffusion processes.Therefore,taking monastery as the carrier,this study establishes a multi-level diffusion model to reconstruct the diffusion networks of Gelugpa monasteries,as well as a framework to explore the detailed features in the spatial diffusion processes of Gelugpa in Tibetan areas of China based on a geodatabase of Gelugpa monastery.The results show that the multi-level diffusion model has a considerable applicability in the reconstruction of the diffusion networks of Gelugpa monasteries.Gelugpa monasteries in the Three Tibetan Inhabited Areas present disparate spatial diffusion processes with diverse diffusion bases,speeds,stages,as well as diffusion regions and centers.A powerful single-center diffusion-centered Gandan Monastery was rapidly formed in U-Tsang.Kham experienced a slower and more varied spatial diffusion process with multiple diffusion systems far apart from each other.The spatial diffusion process of Amdo was the most complex,with the highest diffusion intensity.Amdo possessed the most influential diffusion centers,with different diffusion shapes and diffusion ranges crossing and overlapping with each other.Multiple natural and human factors may contribute to the formation of Gelugpa monasteries.This study contributes to the understanding of the geography of Gelugpa and provides reference to studies on religion diffusion.展开更多
Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample pat...Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X （1） and X （2） are studied. More gener-ally, let E1, E2？（0,∞） and F ？Rd be Borel sets. A necessary condition and a suffcient condition for P{X（1）（E1）∩X（2）（E2）∩F 6=？}〉0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space （R＋× R＋× Rd,ρb）, whereρb is an unsymmetric metric defined in R＋× R＋× Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.展开更多
In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entro...In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entropy with respect to μ under some condition about Γ and the drift b. Moreover, we revisit the exponential convergence of the usual Fisher entropy under the Bakry Emery condition.展开更多
The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chose...The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chosen as the equivalent martingale measure.By the F-S decomposition,the expression of the locally risk minimizing strategy was presented.Finally,the local risk minimization was applied to index tracking and its relationship with tracking error variance (TEV)-minimizing strategy was obtained.展开更多
A class of multi dimensional degenerate diffusion processes X ε(t) in R r(r≥2) are considered and the asymptotic properties of empirical measures are investigated; here X ε(t) saitisfies the stochastic differen...A class of multi dimensional degenerate diffusion processes X ε(t) in R r(r≥2) are considered and the asymptotic properties of empirical measures are investigated; here X ε(t) saitisfies the stochastic differential equation dX ε(t)=σ(X ε(t)) d W(t)+B(X ε(t)) d t+ εσ～(X ε(t)) d W(t),ε>0. X ε(t) are small random perturbations of the degenerate diffusion process X(t), which satisfies the stochastic differential equation dX(t)=σ(X(t)) d W(t)+B(X(t)) d t. A large deviation theorem for projection measures ν on R r-n (n<r) of empirical measures μ are proved展开更多
The cone condition for x to be tegular for B under the elliptic diffusionprocess was proved. We also gave a necessary and sufficient condition for 0 to be regular for thornunder the elliptic diffusion process.
The authors investigate the problem of impulse control of a partially observed diffusion process. The authors study the impulse control of Zakai type equations. The associated value function is characterized as the on...The authors investigate the problem of impulse control of a partially observed diffusion process. The authors study the impulse control of Zakai type equations. The associated value function is characterized as the only viscosity solution of the corresponding quasi-variational inequality. The authors show the optimal cost function for the problem with incomplete information can be approximated by a sequence of value functions of the previous type.展开更多
The grain boundary diffusion process(GBDP)has proven to be an effective method for enhancing the coercivity of sintered Nd-Fe-B magnets.However,the limited diffusion depth and thicker shell struc-ture have impeded the...The grain boundary diffusion process(GBDP)has proven to be an effective method for enhancing the coercivity of sintered Nd-Fe-B magnets.However,the limited diffusion depth and thicker shell struc-ture have impeded the further development of magnetic properties.Currently,the primary debates re-garding the mechanism of GBDP with Tb revolve around the dissolution-solidification mechanism and the atomic substitution mechanism.To clarify this mechanism,the microstructure evolution of sintered Nd-Fe-B magnets during the heating process of GBDP has been systematically studied by quenching at different tem peratures.In this study,it was found that the formation of TbFe_(2) phase is related to the dis-solution of _(2)Fe_(14)B grains during GBDP with Tb.The theory of mixing heat and phase separation further confirms that the Nd_(2)Fe_(14)B phase dissolves to form a mixed phase of Nd and TbFe_(2),which then solidifies into the(Nd,Tb)_(2)Fe_(14)B phase.Based on the discovery of the TbFe_(2) phase,the dissolution-solidification mechanism is considered the primary mechanism for GBDP.This is supported by the elemental content of the two typical core-shell structures observed.展开更多
In this work,the effect of the Al addition amount in the TbAl coatings on the grain boundary diffusion proces s(GBDP)of Tb were systematically explored.Direct current magnetron sputtering(DCMS)method was utilized in c...In this work,the effect of the Al addition amount in the TbAl coatings on the grain boundary diffusion proces s(GBDP)of Tb were systematically explored.Direct current magnetron sputtering(DCMS)method was utilized in co-sputtering manner to synthesize the TbAl coatings with certain Tb consumption and various Al addition amount.Results show that the moderate Al addition amount significantly improves the wettability of grain boundary(GB)phases,thereby acquiring more continuous and uniform Tb-rich shells and GB phases between matrix phases,as well as deeper diffusion depth and denser microstructure.The largest increase amplitude of intrinsic coercivity(Hcj)is improved by 78.4%in TbAIdiffused magnet compared to the pure Tb-diffused magnet,while the remanence(Br)is expected to show an overall decreasing tendency accompanied with a slight increase in the decreasing process.However,when the Al addition amount is excessive,magnetic dilution effect is enhanced,and the Tbrich shells and GB phases between matrix phases become fuzzy and even invisible,which in turn deteriorates the magnetic properties of diffused magnets.展开更多
In this paper,a class of reaction diffusion processes with general reaction rates is studied.A necessary and sufficient condition for the reversibility of this calss of reaction diffusion processes is given,and then t...In this paper,a class of reaction diffusion processes with general reaction rates is studied.A necessary and sufficient condition for the reversibility of this calss of reaction diffusion processes is given,and then the ergodicity of these processes is proved.展开更多
In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusio...In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.展开更多
Two typical ARCH models: the ASDARCH model and the APARCH model are analyzed. Let Y k and σ 2 k denote the log returns and the volatility. When the time interval h goes to zero, (Y k,σ 2 k), as a dis...Two typical ARCH models: the ASDARCH model and the APARCH model are analyzed. Let Y k and σ 2 k denote the log returns and the volatility. When the time interval h goes to zero, (Y k,σ 2 k), as a discrete time Markov chain system, weakly converges to a continuous time diffusion process. The continuous time approximation of the ASDARCH model is done using two different methods. With some transformation, these two results are equivalent to high frequency data. The continuous time approximation of the APARCH model is obtained by a different procedure.展开更多
In the present paper we consider the small random perturbations of one-dimensional diffosion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistic...In the present paper we consider the small random perturbations of one-dimensional diffosion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero.展开更多
We consider the maximum likelihood estimator of the unknown parameter in aclass of nonstationary diffusion processes. We give further a precise estimate for the error of theestimator.
We consider the diffusion process Xt on Rn with radial diffusion and drift coefficients. We prove that once the one-dimensional diffusion ｜Xt｜ has algebraic L2-convergence, so does Xt. And some classical examples ar...We consider the diffusion process Xt on Rn with radial diffusion and drift coefficients. We prove that once the one-dimensional diffusion ｜Xt｜ has algebraic L2-convergence, so does Xt. And some classical examples are discussed.展开更多
For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirich...For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirichlet formεa^(N)(f,f):=∑n=i^N∫△(N)(1-∑1≤i≤N^xi)xn(Эnf)^2(x)μα^(N)(dx)with Domain D(εa^(N))being the closure of C^1(△^(N)).We prove the Nash inequalityμa^(N)(f^2)≤Cεa^(N)(f,f)^p/(p+1)μa^(N)(|f|)^2/(p+1),f∈D(εa^(N)),μa^(N)(f)=0 for some constant C>0 and p=(aN+1-1)++∑i^N=11∨(2ai),where the constant p is sharp when max1≤i≤N ai≤1/2 and aN+1≥1.This Nash inequality also holds for the corresponding Fleming-Viot process.展开更多
We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small para...We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.展开更多
Let τ_D denote the lifetime of a diffusion process on domain DR^d. This paper presents a sufficient condition for the exponential moment of τ_D to be finite. Here, both of the domain and the diffusion operator are g...Let τ_D denote the lifetime of a diffusion process on domain DR^d. This paper presents a sufficient condition for the exponential moment of τ_D to be finite. Here, both of the domain and the diffusion operator are general. As an application, the main result of Gao(1995) for conditioned diffusions is improved on.展开更多
Using time reversal for diffusions and Aroson's estimates, we obtain several results on the compact properties of a conditional diffusion process in a small time interval. In particular, we establish the large dev...Using time reversal for diffusions and Aroson's estimates, we obtain several results on the compact properties of a conditional diffusion process in a small time interval. In particular, we establish the large deviation property for a conditional diffusion process.展开更多
Let E be a separable Banach space and μ be a probability measure on E. We consider Dirichlet forms εon L2(E,m).A special compactification MГ of E is studied in order to give a simple sufficient condition which ensu...Let E be a separable Banach space and μ be a probability measure on E. We consider Dirichlet forms εon L2(E,m).A special compactification MГ of E is studied in order to give a simple sufficient condition which ensures that the complement MГ-E has zero ε-capacity.As an application we prove that the classical Dirichlet forms introduced in Albeverio-Rockner[1]satisfy this sufficient condition.展开更多
基金supported by the Humanities and Social Sciences Foundation of the Ministry of Education of China(Grant No.18YJAZH140).
文摘Gelugpa is the most influential extant religious sect of Tibetan Buddhism,which is the spiritual prop for Tibetans,with thousands of monasteries and followers in Tibetan areas of China.Studies on the spatial diffusion processes of Gelugpa can not only reveal its historical geographical development but also lay the foundation for anticipating its future development trend.However,existing studies on Gelugpa lack geographical perspective,making it difficult to explore the spatial characteristics.Furthermore,the prevailing macro-perspective overlooks spatiotemporal heterogeneity in diffusion processes.Therefore,taking monastery as the carrier,this study establishes a multi-level diffusion model to reconstruct the diffusion networks of Gelugpa monasteries,as well as a framework to explore the detailed features in the spatial diffusion processes of Gelugpa in Tibetan areas of China based on a geodatabase of Gelugpa monastery.The results show that the multi-level diffusion model has a considerable applicability in the reconstruction of the diffusion networks of Gelugpa monasteries.Gelugpa monasteries in the Three Tibetan Inhabited Areas present disparate spatial diffusion processes with diverse diffusion bases,speeds,stages,as well as diffusion regions and centers.A powerful single-center diffusion-centered Gandan Monastery was rapidly formed in U-Tsang.Kham experienced a slower and more varied spatial diffusion process with multiple diffusion systems far apart from each other.The spatial diffusion process of Amdo was the most complex,with the highest diffusion intensity.Amdo possessed the most influential diffusion centers,with different diffusion shapes and diffusion ranges crossing and overlapping with each other.Multiple natural and human factors may contribute to the formation of Gelugpa monasteries.This study contributes to the understanding of the geography of Gelugpa and provides reference to studies on religion diffusion.
基金supported by National Natural Science Foundation of China(11371321)Zhejiang Provincial Natural Science Foundation of China(Y6100663)the Key Research Base of Humanities and Social Sciences of Zhejiang Provincial High Education Talents(Statis-tics of Zhejiang Gongshang University)
文摘Let X（1） = {X（1）（s）, s ∈ R＋} and X（2） = {X（2）（t）, t ∈ R＋} be two inde-pendent nondegenerate diffusion processes with values in Rd. The existence and fractal dimension of intersections of the sample paths of X （1） and X （2） are studied. More gener-ally, let E1, E2？（0,∞） and F ？Rd be Borel sets. A necessary condition and a suffcient condition for P{X（1）（E1）∩X（2）（E2）∩F 6=？}〉0 are proved in terms of the Bessel-Riesz type capacity and Hausdorff measure of E1 × E2 × F in the metric space （R＋× R＋× Rd,ρb）, whereρb is an unsymmetric metric defined in R＋× R＋× Rd. Under reasonable conditions, results resembling those of Browian motion are obtained.
基金Supported by the National Natural Science Foundation of China (11001208)the Fundamental Research Funds for the Central Universities
文摘In this paper, we introduce the definition of Γ-Fisher entropy. For the stochastic differential equation in Rd and the unique invariant probability measure μ, we obtain the exponential convergence of Γ-Fisher entropy with respect to μ under some condition about Γ and the drift b. Moreover, we revisit the exponential convergence of the usual Fisher entropy under the Bakry Emery condition.
基金National Natural Science Foundations of China (No. 11071076,No. 11126124)
文摘The hedging problem for insiders is very important in the financial market.The locally risk minimizing hedging was adopted to solve this problem.Since the market was incomplete,the minimal martingale measure was chosen as the equivalent martingale measure.By the F-S decomposition,the expression of the locally risk minimizing strategy was presented.Finally,the local risk minimization was applied to index tracking and its relationship with tracking error variance (TEV)-minimizing strategy was obtained.
文摘A class of multi dimensional degenerate diffusion processes X ε(t) in R r(r≥2) are considered and the asymptotic properties of empirical measures are investigated; here X ε(t) saitisfies the stochastic differential equation dX ε(t)=σ(X ε(t)) d W(t)+B(X ε(t)) d t+ εσ～(X ε(t)) d W(t),ε>0. X ε(t) are small random perturbations of the degenerate diffusion process X(t), which satisfies the stochastic differential equation dX(t)=σ(X(t)) d W(t)+B(X(t)) d t. A large deviation theorem for projection measures ν on R r-n (n<r) of empirical measures μ are proved
基金Supported by the National Natural Science Foundation of China(201130486) and a Grant from the Ministry of Education of China
文摘The cone condition for x to be tegular for B under the elliptic diffusionprocess was proved. We also gave a necessary and sufficient condition for 0 to be regular for thornunder the elliptic diffusion process.
文摘The authors investigate the problem of impulse control of a partially observed diffusion process. The authors study the impulse control of Zakai type equations. The associated value function is characterized as the only viscosity solution of the corresponding quasi-variational inequality. The authors show the optimal cost function for the problem with incomplete information can be approximated by a sequence of value functions of the previous type.
基金supported by the National Key Research and Development Program of China(2022YFB3505503)the National Natural Science Foundation of China(52201230)+2 种基金the Key R&D Program of Shandong Province(2022CXGC020307)the China Postdoctoral Science Foundation(2022M71204)the Beijing NOVA Program(Z211100002121092).
文摘The grain boundary diffusion process(GBDP)has proven to be an effective method for enhancing the coercivity of sintered Nd-Fe-B magnets.However,the limited diffusion depth and thicker shell struc-ture have impeded the further development of magnetic properties.Currently,the primary debates re-garding the mechanism of GBDP with Tb revolve around the dissolution-solidification mechanism and the atomic substitution mechanism.To clarify this mechanism,the microstructure evolution of sintered Nd-Fe-B magnets during the heating process of GBDP has been systematically studied by quenching at different tem peratures.In this study,it was found that the formation of TbFe_(2) phase is related to the dis-solution of _(2)Fe_(14)B grains during GBDP with Tb.The theory of mixing heat and phase separation further confirms that the Nd_(2)Fe_(14)B phase dissolves to form a mixed phase of Nd and TbFe_(2),which then solidifies into the(Nd,Tb)_(2)Fe_(14)B phase.Based on the discovery of the TbFe_(2) phase,the dissolution-solidification mechanism is considered the primary mechanism for GBDP.This is supported by the elemental content of the two typical core-shell structures observed.
基金Project supported by National Key Research and Development Program of China(2021YFB3500100)National Natural Science Foundation of China(52301068)。
文摘In this work,the effect of the Al addition amount in the TbAl coatings on the grain boundary diffusion proces s(GBDP)of Tb were systematically explored.Direct current magnetron sputtering(DCMS)method was utilized in co-sputtering manner to synthesize the TbAl coatings with certain Tb consumption and various Al addition amount.Results show that the moderate Al addition amount significantly improves the wettability of grain boundary(GB)phases,thereby acquiring more continuous and uniform Tb-rich shells and GB phases between matrix phases,as well as deeper diffusion depth and denser microstructure.The largest increase amplitude of intrinsic coercivity(Hcj)is improved by 78.4%in TbAIdiffused magnet compared to the pure Tb-diffused magnet,while the remanence(Br)is expected to show an overall decreasing tendency accompanied with a slight increase in the decreasing process.However,when the Al addition amount is excessive,magnetic dilution effect is enhanced,and the Tbrich shells and GB phases between matrix phases become fuzzy and even invisible,which in turn deteriorates the magnetic properties of diffused magnets.
基金Ying-Tung Fok Education Foundation and NSFCNSFC and by Anhui Education Commitee..
文摘In this paper,a class of reaction diffusion processes with general reaction rates is studied.A necessary and sufficient condition for the reversibility of this calss of reaction diffusion processes is given,and then the ergodicity of these processes is proved.
基金Supported in part by NSFC(Grant No.11771047)Hu Xiang Gao Ceng Ci Ren Cai Ju Jiao Gong Cheng-Chuang Xin Ren Cai(Grant No.2019RS1057)。
文摘In this paper,we define the generalized diffusion operator L=d/dMd/dS for two suitable measures on the line,which includes the generators of the birth-death processes,the one-dimensional diffusion and the gap diffusion among others.Via the standard resolvent approach,the associated generalized diffusion processes are constructed.
基金Supported by the National Natural Science Foundationof China(No.79970 12 0 )
文摘Two typical ARCH models: the ASDARCH model and the APARCH model are analyzed. Let Y k and σ 2 k denote the log returns and the volatility. When the time interval h goes to zero, (Y k,σ 2 k), as a discrete time Markov chain system, weakly converges to a continuous time diffusion process. The continuous time approximation of the ASDARCH model is done using two different methods. With some transformation, these two results are equivalent to high frequency data. The continuous time approximation of the APARCH model is obtained by a different procedure.
基金the Young Teachers Foundation of Beijing Institute of Technology
文摘In the present paper we consider the small random perturbations of one-dimensional diffosion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero.
基金Research partially supported by N.S.F.Grants DMS-0203823Doctoral Program Foundation of the Ministry of Education of China,Grant No.20020269015
文摘We consider the maximum likelihood estimator of the unknown parameter in aclass of nonstationary diffusion processes. We give further a precise estimate for the error of theestimator.
基金This work was supported by the National Natural Science Foundation of China （Grant Nos. 11101040, 11371283, 11431014）, YETP0264, 985 Projects, and the Fundamental Research Funds for the Central Universities.
文摘We consider the diffusion process Xt on Rn with radial diffusion and drift coefficients. We prove that once the one-dimensional diffusion ｜Xt｜ has algebraic L2-convergence, so does Xt. And some classical examples are discussed.
基金The authors would like to thank the referees for helpful comments on an earlier version of the paper.This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11771326,11726627,11831014).
文摘For any N≥2 andα=(α1…,αN+1)∈(0,∞)^N+1,letμa^(N)be the Dirichlet distribution with parameterαon the set△(N):={x^μa∈[0,1]^N:∑1≤i≤N^xi≤1}.The multivariate Dirichlct diffusion is associated with the Dirichlet formεa^(N)(f,f):=∑n=i^N∫△(N)(1-∑1≤i≤N^xi)xn(Эnf)^2(x)μα^(N)(dx)with Domain D(εa^(N))being the closure of C^1(△^(N)).We prove the Nash inequalityμa^(N)(f^2)≤Cεa^(N)(f,f)^p/(p+1)μa^(N)(|f|)^2/(p+1),f∈D(εa^(N)),μa^(N)(f)=0 for some constant C>0 and p=(aN+1-1)++∑i^N=11∨(2ai),where the constant p is sharp when max1≤i≤N ai≤1/2 and aN+1≥1.This Nash inequality also holds for the corresponding Fleming-Viot process.
基金The authors would like to thank the referees for providing many helpful comments and suggestions. Research of the second author was supported in part by the National Natural Science Foundation of China （Grant No. 11171024）.
文摘We consider the asymptotic property of the diffusion processes with Markovian switching. For a general case, we prove a large deviation principle for empirical measures of switching diffusion processes with small parameters.
基金Research supported in part by NNSFC (19631060) Beijing Nornaal University
文摘Let τ_D denote the lifetime of a diffusion process on domain DR^d. This paper presents a sufficient condition for the exponential moment of τ_D to be finite. Here, both of the domain and the diffusion operator are general. As an application, the main result of Gao(1995) for conditioned diffusions is improved on.
文摘Using time reversal for diffusions and Aroson's estimates, we obtain several results on the compact properties of a conditional diffusion process in a small time interval. In particular, we establish the large deviation property for a conditional diffusion process.
文摘Let E be a separable Banach space and μ be a probability measure on E. We consider Dirichlet forms εon L2(E,m).A special compactification MГ of E is studied in order to give a simple sufficient condition which ensures that the complement MГ-E has zero ε-capacity.As an application we prove that the classical Dirichlet forms introduced in Albeverio-Rockner[1]satisfy this sufficient condition.
