Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

[摘要] This article aims to give a formula for differentiating, with respect toT, an expression of the formλ(T,x):=𝔼x[f(XT)e−∫0TV(Xs)ds(det (I+KX,T))P], wherep≥0andXis a diffusion process starting fromx, taking values in a manifold, and the expectation is takenwith respect to the law of this process.KX,T:L2([0,T)→ℝN)→L2([0,T)→ℝN)is a trace class operator defined byKX,Tf(s)=∫0TH(s⋀t)Γ(X(t))f(t)dt, whereH,Γare locally Lipschitz, positiveN×Nmatrices.

[学科分类] 应用数学