A basic tool for studying the space of loops is a probability measure playing the role of the Lebesgue measure which we hope to construct by conditioning a Brownian motion to return to its initial point at time 1. For this small time estimates for $/log p(t,x,y)$, the logarithmic heat kernel, are crucial. In small time, the heat kernel and its derivatives should behave exactly like the Gaussian kernel. Or does it?  One such set of estimates, crucial for the study of loop spaces, are known to hold only on compact manifolds or under curvature conditions. Can we remove these curvature conditions?A stochastic variation method, found in Bismut's book, blending the Cameron-Martin theorem from Malliavin Calculus with the theory of stochastic differential equations (SDEs) has been used widely and lead to fundamental results. The idea is: varying the initial condition on the initial point of an SDE can be realised by varying the noise by a rotation and a Cameron-Martin translation. It is excellent for working our first order derivative formulas, but with serious limitations.In a work with Chen Xin and Wu Bo, we introduce a new stochastic variation, of second orders, that tick all the wish boxes. As an application we obtain estimates on the second order derivatives of the logarithmic heat kernel, removing all curvature restrictions.We will explain the basic ideas behind all these.

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