Short time asymptotics of random heat kernels (Q2712228)

Consider the stochastic differential equation NEWLINE\[NEWLINEdx=X_0 (x)dt+ \sum^r_{j=1} X_j(x)dB_j (t)+\sum^m_{k=1} Y_k(x)d W_k(t)NEWLINE\]NEWLINE on \(\mathbb{R}^d\), where \((B_j, W_k)_{j,k}\) is a standard Wiener process on \(\mathbb{R}^{r+m}\), and \((X_j)\) and \((Y_k)\) are complete \(C^\infty\) vector fields. Defining a random transition semigroup by taking expectation only with respect to \((B_j)_{1\leq j\leq r}\), the notion of a random heat kernel is introduced. Then conditions on the vector fields are formulated which allow to characterise the short time asymptotics of these random heat kernels in a fashion similar to the author's results on the classical heat kernel. Proofs are announced to be published elsewhere.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].