The positivity of the heat kernel on Heisenberg group (Q2855473)

The (one-dimensional) Heisenberg group \({\mathbf H}_1\) is the nilpotent group structure in \({\mathbb R}^3\) given by the product law NEWLINE\[NEWLINE (x_1,x_2,y)\cdot(x_1',x_2',y')=\big(x_1+x_1',x_2+x_2',y+y'+\frac12(x_2x_1'-x_1x_2')\big). NEWLINE\]NEWLINE The natural sub-Riemannian structure on this Lie group, see [\textit{L. Capogna} et al., An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Basel: Birkhäuser (2007; Zbl 1138.53003)], induces a sub-elliptic operator called sub-Laplacian NEWLINE\[NEWLINE \Delta_{{\mathbf H}_1}=\frac12(X_1^2+X_2^2), NEWLINE\]NEWLINE where \(X_1\) and \(X_2\) are the vector fields given by NEWLINE\[NEWLINE X_1=\frac\partial{\partial x_1}+\frac{x_2}2\frac\partial{\partial y},\quad X_2=\frac\partial{\partial x_2}-\frac{x_1}2\frac\partial{\partial y}. NEWLINE\]NEWLINE It is known, see [\textit{B. Gaveau}, Acta Math. 139, 95--153 (1977; Zbl 0366.22010)], that for \((x,y)=(x_1,x_2,y)\in{\mathbf H}_1\) and \(t>0\), the heat kernel \(P_{{\mathbf H}_1}(t;x,y)\) of the heat equation \(\Delta_{{\mathbf H}_1}-\frac\partial{\partial t}\) is given by the formula NEWLINE\[NEWLINE P_{{\mathbf H}_1}(t;x,y)=\frac1{(2\pi t)^2}\int_C e^{\frac{f(\eta,x,y)}{t}}\frac{\eta}{2\sinh\dfrac{\eta}2}d\eta, NEWLINE\]NEWLINE for some appropriate path of integration \(C\), and NEWLINE\[NEWLINE f(\eta,x,y)=i\eta y-\frac{\eta}4(x_1^2+x_2^2)\coth\frac{\eta}2. NEWLINE\]NEWLINE The aim of the present paper is to give a proof of the fact that \(P_{{\mathbf H}_1}(t;x,y)>0\) for all \(t>0\) and all \((x,y)\in{\mathbb R}^3\), using techniques from complex mechanics instead of the more classical probabilistic approach, see [Gaveau, loc. cit.; \textit{A. Hulanicki}, Stud. Math. 56, 165--173 (1976; Zbl 0336.22007)]. A consequence of this result is the positivity of the heat kernel associated to the Grušin operator NEWLINE\[NEWLINE \frac12\left(\frac{\partial^2}{\partial x^2}+x^2\frac{\partial^2}{\partial y^2}\right) NEWLINE\]NEWLINE in \({\mathbb R}^2\), see [\textit{C.-H. Chang} et al., Bull. Inst. Math., Acad. Sin. (N.S.) 4, No. 2, 119--188 (2009; Zbl 1178.53031)].