All Questions
21 questions
5
votes
0
answers
411
views
Is it really interesting to prove well-posedness of unsolved SPDE?
Lots of nonlinear SPDE remained open for decades (especially the non-deterministic ones in higher dimensions because of the regularity of the noise) until Hairer's breakthrough (regularity structures),...
- 573
2
votes
0
answers
93
views
$\Phi_d^3$ SPDE
One of the first prototypes of a singular stochastic PDE is the $\Phi_d^4$ SPDE
$$\partial_t u=\Delta u-u^3+\xi,$$
where $\xi$ is space-time white noise. It is difficult to study because $u$ is ...
- 1,904
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
0
answers
196
views
Towards Schauder estimates: smoothing effect of the semi-group generated by $\Delta+(-\Delta)^{1/2}$
Consider the semi-group $(P_r)_r$ generated by $\Delta+(-\Delta)^{1/2}:$ for a distribution $f$ let $P_rf:=p(r,\cdot)*f$ where $p(r,x):=\sum_{q \in \mathbb{Z}^d}e^{2\pi\mathrm{i}\langle q,x\rangle}e^{-...
- 573
2
votes
0
answers
136
views
Towards the KPZ: Wiener-Ito integral, Kolmogorov type criterion
Consider a space-time white noise $\xi$ and the heat semi-group $(P_r).$
The following Kolmogorov type criterion allows to construct modifications in Besov Space (Here we have a partition of unity $(\...
- 573
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^...
- 573
4
votes
1
answer
210
views
Local solutions of renormalized stochastic PDE
To illustrate the problem consider the mild formulation of the $\Phi^4_2$ model on $[0,T]\times \mathbb{T}^d$: $$\phi=P_r\phi_0+\int_0^rP_{r-q}(-\phi^3(q))dq+Y_r \ \ \ \ \ \ (1)$$ where $(P_r)_{r \...
- 573
4
votes
0
answers
76
views
Regularity structures-paracontrolled distributions: do they always work for sub-critical SPDE?
Stochastic PDE could be solved using either regularity structures or paracontrolled distributions, as long it's sub-critical.
I was wondering if this was proven, that is every sub-critical SPDE could ...
- 573
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 $b:[0,T]\times V\to H$, $\sigma:[0,T]\times V\to\operatorname{HS}(U_0,H)$, $$V\subseteq H\subseteq V^\ast\...
- 167
1
vote
0
answers
89
views
Heat kernel and estimates
In the article by Hairer-Labbe (A simple construction of the continuum
parabolic Anderson model on $\mathbb{R}^2$), they used the following "well known" fact (picture below) in holder spaces....
- 573
6
votes
0
answers
243
views
Global well posedness of $\phi^4_1$
We consider the $\phi^4_1$ model: $\partial_t\phi=\Delta\phi-\phi^3+\xi$ on $[0,T] \times \mathbb{R},$ where $\xi$ is a space time white noise.
I know how to solve this equation locally on the torus, ...
- 573
1
vote
0
answers
93
views
SPDE via fixed point argument and Young's theorem
Let $(P_r)_{r\geq 0}$ be a strongly continuous semi-group (not necessarily the heat kernel).
It is well known that we can prove local well-posedness of a few SPDE using a fixed point argument: Young's ...
- 573
3
votes
2
answers
403
views
Functional integral formulas for the wave equation and other hyperbolic PDEs
The Feynman–Kac formula provides a functional (Wiener) integral representation of the solution $u$ to the heat equation
\begin{align*}
\partial_t u &= \frac{1}{2}\Delta_x u,\\
u(0,x) &= ...
- 11.8k
2
votes
1
answer
697
views
Reference for representation of heat equation with Neumann boundary condition on smooth domain using reflected Brownian motion
We know that the solution of the heat equation $\partial_tu=\frac 12\Delta u$ with Dirichlet boundary condition $u\rvert_{\partial\Omega}=g$ is $u(t,x)=\mathbb{E}[g(B_\tau)\mid B_t=x]$, with $\tau$ ...
- 715
2
votes
1
answer
136
views
Does higher volatility of SDE imply lower probability of staying positive?
Given two SDEs $X^1$, $X^2$ :
$$X^i_t=1+t+\int_0^t\sigma_i(s)dW_s,\quad \forall t\ge 0,$$
where $\sigma_i:\mathbb R_+\to [1/2,1]$ are non-decreasing s.t. $\sigma_1(t)\le \sigma_2(t)$ for all $t\ge 0$....
- 1,334
3
votes
0
answers
145
views
Density of invariant measure of stochastic differential equation
I have a question: is it possible that an SDE has a "nice" density, but its invariant measure does not have a "nice" density? I asked this question at math.stackexchange but ...
- 931
2
votes
0
answers
95
views
Itō formula for the solution of a SPDE in the distributional sense
Let
$d\in\mathbb N$
$\Lambda\subseteq\mathbb R^d$ be open
$(\Omega,\mathcal A,\operatorname P)$ be a probability space
$(Y_t)_{t\ge0}$ be an $L^2(\Lambda)$-valued process on $(\Omega,\mathcal A,\...
- 167
1
vote
1
answer
508
views
Divergence form degenerate pde and Feynman Kac
Consider
$$ Au:=\operatorname{div}\left(y^{\beta}\nabla u\right) \text{ for } (x,y)\in \mathbb{H} $$
and $u|_{\mathbb{R}}(x,0)=\phi(x)$ and some $\beta\in (0,1)$. For $\phi\in L^{2}(\mathbb{R},dx)$ (...
- 5,474
5
votes
3
answers
1k
views
PDE-oriented textbook on probability and random processes?
I was trained in reaction-diffusion (parabolic/elliptic) PDEs, and my research now focuses on applied optimal tranport. I'd like to learn probability and stochastic processes, mostly their connection ...
- 5,380
4
votes
1
answer
391
views
On Brownian motions
I have a question about Brownian motions and its heat kernel.
Using Dirichlet form theory, we can construct Brownian motions on manifolds, domains of Euclidean space under mild assumptions. For ...
- 721
11
votes
1
answer
2k
views
Feynman-Kac theorem: probabilistic proof of existence of solution to parabolic PDE
Friedman (in his book: PDEs of Parabolic Type) shows how to construct a solution to the Cauchy problem
$$
\partial_t u(t,x) = b(x) \partial_x u(t,x) + \frac{1}{2} \sigma(x)^2 \partial_{x,x} u(t,x)
$$
...
- 271