Details

Autor(en) / Beteiligte

• Kunita, Hiroshi

Titel

Smooth Densities and Heat Kernels

Ist Teil von

• Stochastic Flows and Jump-Diffusions, 2019, Vol.92, p.245-302

Ort / Verlag

Singapore: Springer

Erscheinungsjahr

2019

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• We discuss the existence of the smooth density of ‘nondegenerate’ diffusions and jump-diffusions determined by SDEs. We will use the Malliavin calculus studied in the previous chapter. In Sects. 6.1, 6.2, and 6.3, we consider diffusion processes. It will be shown that any solution of a continuous SDE defined in Chap. 10.1007/978-981-13-3801-4_3 is infinitely H-differentiable. Further, if its generator A(t) is elliptic (or hypo-elliptic), its transition probability Ps,t(x, E) has a density ps,t(x, y), which is a rapidly decreasing C∞-function of x and y. We will show further that the weighted transition function has also a rapidly decreasing C∞-function of x and y, and it is the heat kernel of the backward heat equation associated with the operator Ac(t), which were defined in Chap. 10.1007/978-981-13-3801-4_4. For the proof of these facts, we will apply the Malliavin calculus on the Wiener space discussed in Sects. 10.1007/978-981-13-3801-4_5#Sec1, 10.1007/978-981-13-3801-4_5#Sec2, and 10.1007/978-981-13-3801-4_5#Sec3. In Sects. 6.4, 6.5, and 6.6, we will study jump-diffusions. We will show that if the generator of the jump-diffusion is ‘pseudo-elliptic’, then its weighted transition function has a smooth density. For the proof, we will apply the Malliavin calculus on the Wiener-Poisson space discussed in Sects. 10.1007/978-981-13-3801-4_5#Sec9, 10.1007/978-981-13-3801-4_5#Sec10, and 10.1007/978-981-13-3801-4_5#Sec11. In Sects. 6.7 and 6.8, we discuss short-time estimates of heat kernels, applying the Malliavin calculus again. These facts will be applied to two problems. In Sect. 6.9, we consider the solution of SDEs with big jumps, for which the Malliavin calculus cannot be applied. Instead, we take a method of perturbation and show that the perturbation preserves the smooth density. In Sect. 6.10, we show the existence of the smooth density of the laws of the killed elliptic diffusion or the killed pseudo-elliptic jump-diffusions. The density should be the heat kernel of the backward heat equation on the domain with the Dirichlet boundary condition.