### Further exponential generalization of Pitman's $2M-X$ theorem.

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These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof of the Chern–Gauss–Bonnet theorem.

We consider exponential functionals of a brownian motion with drift in ℝ, defined via a collection of linear functionals. We give a characterisation of the Laplace transform of their joint law as the unique bounded solution, up to a constant factor, to a Schrödinger-type partial differential equation. We derive a similar equation for the probability density. We then characterise all diffusions which can be interpreted as having the law of the brownian motion with drift conditioned on the law of...

In this paper we study upper bounds for the density of solution to stochastic differential equations driven by a fractional Brownian motion with Hurst parameter $Hgt;1/3$. We show that under some geometric conditions, in the regular case $Hgt;1/2$, the density of the solution satisfies the log-Sobolev inequality, the Gaussian concentration inequality and admits an upper Gaussian bound. In the rough case $Hgt;1/3$ and under the same geometric conditions, we show that the density of the solution is smooth and admits an upper...

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