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Let $f$ be a continuous real-valued function on $\mathbb{R}^n$. It is well known that the following are equivalent: $f$ is harmonic. $f$ satisfies the ball mean value property $$ f(x)=\frac{1}{|B(x,r)...

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 ...

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...

I am currently looking at a $1$D-Burger's equation defined by \begin{equation} \label{ex burgers} \left\{ \begin{array}{ll} {} & \frac{\partial V_m}{\partial t} (t,x) = \frac{\sigma^2}{2} \...