# Tagged Questions

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### Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
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### Solution of $\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)$

Consider the PDE $$\frac{\partial f}{\partial t}(t,x) = 2 \frac{\partial^2f}{\partial x^2}(t,x)\tag{1}$$ with $t\ge0,\ x\in\mathbb R,\ f(0,x)=e^x$. I want to find $f(t,x)$. I know that the heat ...
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### Definition of regular point of a boundary with planar brownian motion

This is an exercise in G.Lawler's book Conformally invariant processes in the plane. First he defined regular point of a boundary using brownian motion: Suppose $D$ is a domain in $\mathbb{C}$ with ...
Let $\left(X_{t},\, t\geq0\right)$ be the weak solution to the SDE below with $\alpha,\,\beta,\,\gamma$ constants: $$dX_{t}=(-\alpha X_{t}+\gamma)dt+\beta dB_{t}\quad\forall t\geq0,\, X_{0}=x_{0}$$ ...
### Second order linear partial differential equation: $\partial_t u(t,x)+\frac12 \partial_{x,x} u(t,x)+u(t,x)v(x)=0$
Is there a way to solve $$\partial_t u(t,x)+\frac12 \partial_{x,x}u(t,x)+u(t,x)v(x)=0?$$ This appeared as a condition for $$X_t=u(t,B_t)e^{\int_0^tv(B_s)ds}$$ to be a martingale. With $B$ a ...