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4 votes
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Finiteness of the moments of the Malliavin derivative of the stochastic heat equation

I am studying section 2.4.2 from Nualart's book "The Malliavin calculus and related topics" on the stochastic heat equation. I have some questions on the validity of some estimates for the ...
user574579's user avatar
4 votes
0 answers
113 views

SPDE Renormalization

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 ...
mathex's user avatar
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3 votes
0 answers
80 views

Norm estimate for parabolic SPDE solution

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-\|...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
42 views

Diffusions vs elliptic operators with dkp coefficients

I am wondering if there is any literature on the relationship between diffusions and elliptic equations. In particular I am interested in literature concerning operators with Dahlberg–Kenig–Pipher ...
Diesirae92's user avatar
2 votes
0 answers
89 views

Malliavin calculus for the regularity of the density of the supremum of a process

I am reading Chapter 2 from Nualart's book 'The Malliavin calculus and related topics'. Proposition 2.1.10 gives the conditions for the law of the supremum of a process to have a density. Condition (...
user574579's user avatar
2 votes
0 answers
203 views

Time reversal of infinite-dimensional SDE

Consider the SDE $${\rm d}X_t=b(t,X_t) \, {\rm d}t+\sigma(t,X_t) \, {\rm d}W_t,\tag1$$ where $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
95 views

Itō formula for the solution of a SPDE in the distributional sense

Let $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be open $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
0xbadf00d's user avatar
  • 167
2 votes
0 answers
120 views

Taking limits in stochastic partial differential initial value problems

Background: A (stochastic) Cauchy problem I am interested in looks like this: $$ (1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
Mark's user avatar
  • 657
2 votes
0 answers
135 views

Connection between deterministic and stochastic problems in PDEs

In the study of conservation laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)): Deterministic Cauchy problem: $$(1) \hspace{1cm} \...
Mark's user avatar
  • 657
1 vote
0 answers
106 views

Domain of a reflected stochastic differential equation

I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
fast_and_fourier's user avatar
1 vote
0 answers
124 views

Derive a SPDE of evolutionary type for $u$ from ${\rm d}X(t)=u(t,X(t)){\rm d}t+\xi(t,X(t)){\rm d}W(t)$

Let $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota:U\to V$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ $(\Omega,\mathcal A,\...
0xbadf00d's user avatar
  • 167
0 votes
0 answers
233 views

Probability that d-Brownian Motion ,$d\geq 3$, avoids a fixed set A

In other words, the probability that Brownian motion stays within $A^{c}$. What about for connected and fixed compact sets ? Would that involve solving a heat equation? How can I condition it, so ...
Thomas Kojar's user avatar
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