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If we consider a nice Ornstein-Uhlenbeck process $d x (t) = - \gamma x(t) \,dt + \sigma \,d w (t)$ with $x(0) = x_0 \in (-L,L)$. Here $\gamma, \sigma$ are positive constants and $w(t)$ is a Wiener pro …

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions: Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 …

It is known that for a real valued stochastic process $X_t$ satisfying $$ d X_t = b(X_t) d t + \sigma d W_t $$ where $W$ is real valued Wiener process, the equation for the most probable path from …

Let $\{ X_t, t \geq 0 \}$ be a $\mathbb{R}^d$-valued stochastic process on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assumption $X_t$ is a regenerative process in the sense of https …

Here $q$ is the position and $p$ is the velocity. Let me take $a = \sqrt{2}$ and define $H(q,p) = \frac{q^2+p^2}{2}$. case 1 - b is a constant. We agree on the fact that the generator $L$ of $(q(t) …