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I am wondering if the Poincare inequality holds for the Brownian path space. As the simplest example, let $\{w_t, t \in [0, 1]\}$ be a 1-d standard BM: has independent increments and continuous path …

I check it with the standard Garlerkin method and confirmed that it is right, in both cases (i) and (ii). Discribe the proof briefly (under (ii)): Take a CONS of $H$ as $h_1 = 1$ and $h_k(x) = \co …

Let $X_i$ be a sequence of iid random variables, $E [X] = 0$, $E [X^2] = 1$ and $E [|X|^k] < \infty$ for some $k \ge 3$. Classical local CLT says that the density function $f_n$ of $\frac1{\sqrt n}\s …

Let us consider a generalized O-U process $X_t \in L^2[0, 1]$ defined by the following spde: $dX_t = \frac{1}{2}\partial_x^2X_t + dW_t, $ $\partial_x X_t(0) = \partial_x X_t(1) = 0, $ $X_0 = 0, $ …

Consider a simple SPDE as follows: $\partial_t u(t,x)=\partial_x^2 u(t,x)+V(u(t,x))+\dot{W}(t,x)$, $t>0$, $x\in(0,1)$, $u(t,0)=u(t,1)=0$, $u(0,x)=v(x)$, where $V$ is a bounded, smooth potentia …

Consider a 1-dimensional stochastic heat equation on $[0, 1]$, with boundary conditions of Neumann's type: \begin{equation}\left\{ \begin{aligned} &\partial_t u(t, x) = \frac{1}{2}\partial_x^2 u(t, x …

When reading the paper E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374 I found an argument like the following. Given an bounded and self-adjoint linear operator $K: …

Denote $E = C([0, 1])$. I am consider a 1-dimensional stochastic heat equation on $h$: $$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), \quad\text{ for all } (t, x) \in (0, \ …