To characterize the Neumann problem for nonlinear Fokker-Planck equations,we investigate distribution dependent reflecting stochastic differential equations(DDRSDEs)in a domain.We first prove the well-posedness and es...To characterize the Neumann problem for nonlinear Fokker-Planck equations,we investigate distribution dependent reflecting stochastic differential equations(DDRSDEs)in a domain.We first prove the well-posedness and establish functional inequalities for reflecting stochastic differential equations with singular drifts,and then extend these results to DDRSDEs with singular or monotone coefficients,for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting stochastic differential equations is established.展开更多
By using a split argument due to[1],the transportation cost inequality is established on the free path space of Markov processes.The general result is applied to stochastic reaction diffusion equations with random ini...By using a split argument due to[1],the transportation cost inequality is established on the free path space of Markov processes.The general result is applied to stochastic reaction diffusion equations with random initial values.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
Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications,distribution dependent stochastic differential equations(DDSDEs)have been intensively investigated.In this paper,we summar...Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications,distribution dependent stochastic differential equations(DDSDEs)have been intensively investigated.In this paper,we summarize some recent progresses in the study of DDSDEs,which include the correspondence of weak solutions and nonlinear Fokker-Planck equations,the well-posedness,regularity estimates,exponential ergodicity,long time large deviations,and comparison theorems.展开更多
基金supported by the National Key R&D Program of China(Grant No.2020YFA0712900)National Natural Science Foundation of China(Grant Nos.11831014 and 11921001)。
文摘To characterize the Neumann problem for nonlinear Fokker-Planck equations,we investigate distribution dependent reflecting stochastic differential equations(DDRSDEs)in a domain.We first prove the well-posedness and establish functional inequalities for reflecting stochastic differential equations with singular drifts,and then extend these results to DDRSDEs with singular or monotone coefficients,for which a general criterion deducing the well-posedness of DDRSDEs from that of reflecting stochastic differential equations is established.
基金supported by National Natural Science Foundation of China(11671372,11771326,11831014).
文摘By using a split argument due to[1],the transportation cost inequality is established on the free path space of Markov processes.The general result is applied to stochastic reaction diffusion equations with random initial values.
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This work was supported in part by the National Natural Science Foundation of China(Grant Nos.11771326,11831014,11921001,11801406).
文摘Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications,distribution dependent stochastic differential equations(DDSDEs)have been intensively investigated.In this paper,we summarize some recent progresses in the study of DDSDEs,which include the correspondence of weak solutions and nonlinear Fokker-Planck equations,the well-posedness,regularity estimates,exponential ergodicity,long time large deviations,and comparison theorems.
