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Let $\xi$ be a space-time white noise, that is a centered Gaussian process with covariance $E[\xi_{f}\xi_h]=\int_{\mathbb{R}_+ \times \mathbb{R}^d}fh,$ for $f,h\in L^2(\mathbb{R}_+ \times \mathbb{R}^d …

Let $(\xi_\phi)_{\phi \in L^2(\mathbb{R}_+ \times \mathbb{R}^d,\lambda_d)}$ be a collection of centered Gaussian processes on a probability space $(\Omega,\mathcal{F},P)$ such that $$\forall \phi \in …

Consider a probability space $([0,1],\mathcal{B}([0,1],\lambda),p>1$ and $f \in L^p(]0,\infty[).$ We want to prove Hardy's inequality using martingale theory and Doob's maximal inequalities. Let $\mat …

I am looking for a reference that gives a detailed proof of Chung's law of the iterated logarithm for Brownian motion: $$\liminf_{u\to +\infty}\sqrt{\frac{\ln(\ln(u))}{u}}\sup_{r \in [0,u]}|X_r|=\frac …

Let $f,h,g$ be continuous functions and $B$ a real Brownian motion. We suppose that a.s. $$\forall u \in \mathbb{R}_+,f(B_u)=f(B_0)+\int_0^ug(B_r)dB_r+\frac{1}{2}\int_0^uh(B_r)dr.$$ Prove that $f$ is …

Consider a submartingale $X,$ then for almost every $\omega \in \Omega,$ for every $v \in \mathbb{R},\lim_{u \in \mathbb{{Q},u \uparrow v}}X_u(\omega)$ exist in $\mathbb{R}.$ I was wondering if there …

I am looking for the following article: Al-Hussaini, A. N. A projective limit view of $L_1$-bounded martingales. Rev. Roumaine Math. Pures Appl.26 (1981), no.1, 51–54, but I can't find it anywhere. Do …

Consider the following theorem (picture below) taken from Pardoux's lecture notes: Stochastic partial differential equations available at scholar google: https://scholar.google.ca/scholar?q=etienne+pa …

some SPDE (in higher dimensions) can only be interpreted in a "renormalised" sense. For example considering $\Phi_2^4$ on $\mathbb{R}_+\times \mathbb{T}^d$ the solution is defined as the limit of $\ph …

This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely. Again, we are interested in the local behavior of the $\Phi_2 …

I am looking for a reference with a proof for the following fact: If a right-continuous martingale $(X_r)_{ r \geq 0}$ is such that $X_0=0,(X^2_r-r)_r,(X_r^3-3rX_r)_r,(X_r^4-6rX_r^2+3r^2)_r$ are marti …

Let $\mathbb{T}^d$ be the $d$-dimensional torus, $\mathscr{S}:=C^\infty(\mathbb{T}^d)$ the Schwartz space, $\mathscr{S}'$ the space of tempered distributions. We consider a dyadic partition of unity $ …

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u …