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I am working with the porous medium equation, which I am treating it as a type of Fokker-Planck equation given by: $ \frac{\partial u}{\partial t} = \Delta(u^m), \quad m > 1 $ For this equation, ...

When $X$ satisfies $${\rm d}X_t=\varphi_t{\rm d}t+\Phi_t{\rm d}W_t$$ on a Hilbert space $H$, where $W$ is a $Q$-Wiener process on a Hilbert space $U$, we know by the Ito formula that $$\|X_t\|_H^2-\|...

In the article by Hairer-Labbe (A simple construction of the continuum parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....

We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...

Consider the Kolmogorov forward equation for a Langevin dynamic: $$\DeclareMathOperator{\Div}{div} \begin{cases} \dfrac{\partial}{\partial t} f = \Delta f + \Div(f\nabla V)\\ \\ \displaystyle\int_{\...

I have a dynamic programming problem with an underlying diffusion $$ d X_t = \mu \, dt + d b_t$$ where $b_t$ is a standard brownian motion. The HJB equation for the value function $v(x,t)$ I get is ...