Fundamental solution of kinetic Fokker-Planck operator with anisotropic nonlocal dissipativity (Q2878998)

This work studies the stochastic Hamiltonian system given by \(dX_t=b_1(X_t,\dot{X}_t)dt, \, X_0=x\) and \(d\dot{X}_t=b_2(X_t,\dot{X}_t)dt+dL_t,\,\dot{X}_0=v\). Here, \(b=(b_1,b_2)\) is a vector field on the phase space and \(L_t\) is a \(d\)-dimensional Lévy process with the form of a subordinated Brownian motion \(L_t:= W_{S_t}= (W^1_{S_t^1}, \ldots, W^d_{S_t^d})\), where \(W\) is a \(d\)-dimensional Brownian motion and \(S_t=(S_t^1,\ldots,S_t^d)\) is an independent \(d\)-dimensional \(\mathbb{R}^d_+\)-valued Lévy process, each component \(S^i\) being a subordinator. Under some assumptions on the Lévy measure of \(S\) and on the drift term \(b\) (allowing \(b\) to be of cubic growth), it is proved that there exists a smooth density to the Hamiltonian system which is a fundamental solution to the associated kinetic Fokker-Planck equation with anisotropic nonlocal dissipativity. The main tool is Malliavin calculus, which entails calculating a Malliavin covariance matrix.