The Fundamental Solution of a Degenerate Partial Differential Equation of Parabolic Type

Linear and Nonlinear Ultraparabolic Equations of Kolmogorov Type Arising in Diffusion Theory and in Finance

“…Theorem 4.1, which in particular improves and generalizes the previous results by Weber [48], Il'in [23] and Sonin [47], was proved in [40] by adapting the Levi's parametrix method to the Lie group and metric structures related to the matrix B (cf. Section 2).…”

“…Weber [48] in 1951, Il'in [23] in 1964 and Sonin [47] in 1967 constructed a fundamental solution for linear Kolmogorov operators. Regularity results of solutions and first boundary value problems were investigated by Genčev [21],Šatyro [46], Eidelman, Ivasyshen and Malytska [17].…”

Pointwise estimates for a class of non-homogeneous Kolmogorov equations

“…A systematic study of this class of operators, when the coefficents a i j are constant, has been carried out by Kupcov [18], and by Lanconelli and Polidoro [19]. The existence of a fundamental solution has been proved by Weber [36], Il'in [17], Eidelman [12] and Polidoro [29,30] in the case of Hölder continuous coefficients a i j . Pointwise upper and lower bound for the fundamental solution, mean value formulas and Harnack inequalities are given in [29,30]; Schauder type estimates have been proved by Satyro [35], Lunardi [22], Manfredini [23].…”

Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type

“…As a consequence of Lemma 3.1 we have the following monotonicity property of Kolmogorov averages. Er (X o ' w o ' to;X,w,t) Xh Ah(\7(y,wlh(X O ' wo' to;X,w,t))· \7 (y,wl H(XO' wo' to; X, w, t) = --;r;…”