Fredholm Determinant of an Integral Operator Driven by a Diffusion Process

This article aims to give a formula for differentiating, with respect to T, an expression of the form λ(T,x):=ð”¼x[f(XT)e−∫0TV(Xs)ds(detâ¡(I+KX,T))P], where p≥0 and X is a diffusion process starting from x, taking values in a manifold, and the expectation is taken with respect to the law of this process. KX,T:L2([0,T)→â„ÂN)→L2([0,T)→â„ÂN) is a trace class operator defined by KX,Tf(s)=∫0TH(s∧t)Γ(X(t))f(t)dt, where H, Γ are locally Lipschitz, positive N×N matrices