All Questions
19 questions
4
votes
0
answers
122
views
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 ...
- 61
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 ...
- 573
3
votes
2
answers
403
views
Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
- 1,904
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 ...
- 63.9k
2
votes
1
answer
207
views
Elliptic PDEs in BSDEs and in optimal control
This soft/reference question is related to this MO post of a similar nature.
What are some examples of elliptic PDEs appearing in control and BSDEs?
- 6,205
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 (...
- 61
5
votes
1
answer
205
views
Continuity dependence and convergence of the renormalized $\Phi^4_2$ model
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^...
- 10.3k
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 $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 1
3
votes
0
answers
80
views
Norm estimate for parabolic SPDE solution
When $X$ satisfies $${\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-\|...
- 167
3
votes
1
answer
174
views
Stochastic representation of Laplace equation with Neumann boundary condition
Consider nice domain $D\subset \mathbb R^d$ and $\Delta u =0$ with $u\big|_{\partial D}=g$. It is well known that $u(x)=E^x[g(B(\tau))]$ where $\tau$ is exit time of $B$ from the domain $D$.
What if ...
- 300
11
votes
1
answer
2k
views
Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
- 6,367
3
votes
3
answers
348
views
Is $p_t(y,x)$ the kernel of the time reversal of the diffusion $X$, for $p_t(x,y)$ the kernel of $X$?
Short version. If $X$ is a diffusion with generator $L$ and the Lebesgue measure is invariant for $X$, then $L^*$ has no term of order zero and it corresponds to another diffusion $X^*$. Denoting by $...
- 3,669
5
votes
1
answer
445
views
Schwartz regularity for the density of a stochastic process
Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables
$$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$
It ...
- 3,669
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,\...
- 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:
$$
(1) \hspace{0.5cm} \frac{\partial u}{\partial t}+A(u) \cdot \frac{\partial u}{\partial x} =\nu \cdot \frac{\partial^2 ...
- 657
1
vote
0
answers
106
views
Domain of a reflected stochastic differential equation
I am currently investigating the domain of the infinitesimal generator of a reflected stochastic differential equation (for a smooth and bounded domain) with Lipschitz coefficients. Namely SDEs of the ...
- 63.9k
2
votes
0
answers
135
views
Connection between deterministic and stochastic problems in PDEs
In the study of conservation laws in partial differential equations relatively often we see this two problems (problem (1) more than problem (2)):
Deterministic Cauchy problem:
$$(1) \hspace{1cm} \...
- 657
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 $U$
$(\Omega,\mathcal A,\...
- 167
0
votes
0
answers
233
views
Probability that d-Brownian Motion ,$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 ...
- 5,474