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Let $v(x, t) = \mathbb E [f(x + W_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$: $$\partial_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W_t) W_t ].$$ I am ...

I am looking for a super(sub) harmonic function for an elliptic operator. Let $n$ be a positive integer. We denote by $(\cdot,\cdot)$ and $|\cdot|$ the standard inner product and norm on $\mathbb{R}^n$...

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...