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.
