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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),...
mathex's user avatar
  • 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 ...
user479223's user avatar
  • 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 ...
mathex's user avatar
  • 573
5 votes
1 answer
139 views

Dispersion of random walk with scaled step sizes

Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$. We define the ...
felipeh's user avatar
  • 452
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^{-...
mathex's user avatar
  • 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 $(\...
mathex's user avatar
  • 573
1 vote
0 answers
48 views

Rigorous analysis of phase transitions and universality in a non-linear model of interacting oscillators

Consider a system of interacting non-linear oscillators governed by the McKean-Vlasov equation: $$\frac{\partial p(x,t)}{\partial t} = \frac{\partial}{\partial x}\left[\frac{\partial V(x)}{\partial x}...
user avatar
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^...
mathex's user avatar
  • 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 \...
mathex's user avatar
  • 573
2 votes
0 answers
137 views

Holder-Besov space and time continuity

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 $(...
mathex's user avatar
  • 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 ...
mathex's user avatar
  • 573
3 votes
1 answer
251 views

Feynman–Kac formula for other operators

I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$ where $x \in \Omega$ and $t > 0$, then $e^{t\Delta_D}f(x) = ...
Emmie's user avatar
  • 41
7 votes
2 answers
307 views

PDE for the probability of Brownian motion staying in an area (reference request)

I am looking for a (preferably some monograph) reference on the following fact: $$ u ( t, x ) = \mathbb{P} \{ x + B_s \in A \ \text{for all} \ s \leq t \} $$ satisfies the heat equation $$ \frac{\...
tsnao's user avatar
  • 620
2 votes
1 answer
131 views

Gradient flows and particle representations

I was looking into gradient flows and their particle representations, mostly in the context of probability. A simple example of this is the continuity equation. Consider evolving a sample $x \sim \...
CComp's user avatar
  • 123
5 votes
1 answer
437 views

Elliptic PDEs in Finance

In mathematical finance, one often encounters parabolic PDEs typically through the Feynman-Kac representation theorem/formula. However, I'm curious are there interesting examples of Elliptic boundary ...
ABIM's user avatar
  • 5,405
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
0 votes
1 answer
54 views

How is this interpolating curve well-defined in the minimizing movement scheme?

Let $\Omega$ be a compact domain of $\mathbb R^d$. Let $\mathcal P (\Omega)$ be the space of probability measures on $\Omega$. For each $\tau >0$, let $(\varrho^\tau_{(k)})_{k \in \mathbb N} \...
Akira's user avatar
  • 835
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....
mathex's user avatar
  • 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, ...
mathex's user avatar
  • 573
2 votes
0 answers
70 views

Reference request : A SPDE model

Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider ...
Fawen90's user avatar
  • 1,399
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 ...
mathex's user avatar
  • 573
1 vote
1 answer
433 views

Derivative of Wasserstein distance $W^p_p$ along solutions of the continuity equation (contradicting statements in different sources)

Let $(\rho^{(i)}_t,{\bf v}^{(i)}_t)$ for $i = 1,2$ be two solutions of the continuity equation $$\partial \rho^{(i)}_t + \nabla\cdot \left({\bf v}^{(i)}_t \rho^{(i)}_t\right) = 0 \label{1}\tag{1}$$ on ...
Fei Cao's user avatar
  • 730
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) &= ...
Emily's user avatar
  • 11.8k
2 votes
1 answer
228 views

When is a stationary measure of a Markov chain "exponentially localized"?

Here exponentially localized can be thought in a non-rigorous manner as a measure that is mostly supported on a sparse number of nodes. Some intuition can gained by thinking about a diffusion process, ...
Piyush Grover's user avatar
2 votes
0 answers
94 views

Convergence of Green's function of fractional heat equation

For the fractional heat equation \begin{equation} \partial_t u + (-\Delta^s)u=0 \text{ in } \mathbb{R}^d \times (0,\infty), \end{equation} where $s \in (0,1)$ where the fractional laplacian is the ...
Eyumi's user avatar
  • 21
2 votes
1 answer
483 views

What is an analogous version of the Ornstein–Uhlenbeck process on Riemannian manifolds?

Recall that the Ornstein–Uhlenbeck (OU) process in $\mathbb{R}^d$ is defined by the following SDE, $$ d Z_t=\frac{-1}{2} Z_t d t+d W_t, \quad t \geqslant 0 $$ where $\left(W_t\right)_{t \geqslant 0}$ ...
Student's user avatar
  • 537
1 vote
1 answer
386 views

Role of verification theorems in stochastic optimal control?

I am currently working on the optimal control of certain classes of stochastic processes and I have difficulties understanding the roles of verification theorems. My problem is the following: I am not ...
KBS's user avatar
  • 23
0 votes
0 answers
34 views

Inequalities for generalized variance

Let $(X, \mu)$ be a measured space with $\mu(X) = 1$. Given $\phi \in L^\infty(X, \mu)$, $\phi > 0$, let me define, for $\alpha \geq 1$, $\beta > 0$, the quantity $$ I(\alpha, \beta) = \left(\...
Romain Gicquaud's user avatar
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 $\tau$ ...
MikeG's user avatar
  • 715
1 vote
0 answers
166 views

Wiener Integral and its distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space. Let $(W(t))_{x \in \mathbb{R}^d}$ be a Gaussian random field. Then, we can define Wiener integral $\int_{\mathbb{R}^d} f(\xi) \, dW(\xi)$...
heppoko_taroh's user avatar
4 votes
0 answers
134 views

Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that $$ \Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu $$ where the entropy $$ \Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...
leo monsaingeon's user avatar
3 votes
1 answer
466 views

Equivalence between two fractional Sobolev spaces

For $s \in (0,1)$, we consider the spectral fractional Laplacian \begin{align} (-\Delta)^{-s}u = \sum_{k=1}^{\infty}\lambda_k^{-s}(\phi_k,u)_{L^2}\phi_k \end{align} where \begin{align*} \begin{cases} ...
Zac's user avatar
  • 161
1 vote
0 answers
152 views

Poisson Kernel and solution formula for fractional elliptic problem

$$ k (-\Delta)^s u + u = 0, \qquad x \in U, \\ u(x) = f(x), \qquad x \in \mathbb R^n \setminus U, $$ with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
Riku's user avatar
  • 839
3 votes
0 answers
158 views

$L^\infty-L^\infty$ bounds for heat semigroups constructed from the Dirichlet Laplacian

Let $D \subset \mathbb{R}^n$ be a bounded domain with Lipschitz boundary, and let $\Delta$ be the Laplace operator with the Dirichlet boundary condition on $D$. Let $e^{t\Delta}$ be the corresponding ...
SMS's user avatar
  • 1,407
1 vote
0 answers
248 views

Regularity of Fokker-Planck equation

Consider solutions $\rho_{1,2}$ of the Fokker-Planck equation $$\begin{cases}\partial_t \rho_i = \Delta \rho_i + \nabla \cdot (\rho_i \nabla \Phi_{1,2})\\ \rho_i(0,\cdot) = \rho^0 \end{cases}$$ for ...
Peter Koepernik's user avatar
2 votes
0 answers
138 views

Update on Viskov's paper on random processes, Lagrange inversion, and the Heisenberg–Weyl algebra

"A Random Walk with a Skip-Free Component and the Lagrange Inversion Formula" by Viskov presents connections among Lagrange inversion and measures of random Lévy processes. The freely ...
Tom Copeland's user avatar
  • 10.5k
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 $t\ge 0$....
GJC20's user avatar
  • 1,334
0 votes
1 answer
162 views

Iterated integrations by parts using the fractional Laplacian

Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that $$\...
Riku's user avatar
  • 839
2 votes
0 answers
86 views

Continuity of the entropy of the solution of a parabolic PDE at $t=0$

Consider the following initial value problem for a parabolic PDE : $$\begin{cases} \textrm{div}\big(A\,\nabla u(t,x) + b(x)\, u(t,x)\big) \,=\,\partial_t u(t,x) \quad x\in\mathbb R^d\,,\ t>0 \\[4pt]...
tituf's user avatar
  • 311
2 votes
1 answer
268 views

Existence of the derivative of functionals of Brownian motion

Let $v(x, t) = \mathbb E [f(x + W_t)]$ with a Brownian motion $W$. Then, Malliavin calculus leads to the sensitivity in $x$: $$\partial_x v(x, t) = \frac{1}{t} \mathbb E [ f(x + W_t) W_t ].$$ I am ...
kenneth's user avatar
  • 1,399
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 ...
Oleg's user avatar
  • 931
0 votes
0 answers
173 views

Lemma 3.10 of paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain

I am reading a paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain. And I have a questions in the proof of lemma 3.10. Please click the paper title for the link. The ...
Lev Bahn's user avatar
  • 239
2 votes
1 answer
171 views

Mean value formula for fractional heat equation

For the solution $u(z) = u(t,x)$ of the heat equation $u_t -\Delta u = 0$ we have $$u(z_0) = \int_{\Omega_r(z_0)}u(z) K_r(z_0-z) dz,$$ where $$\Omega_r(z_0) = \left\{z \in \mathbb{R}^{N+1}: \Gamma(z_0-...
Zac's user avatar
  • 161
4 votes
1 answer
387 views

Asymptotic formula for fractional Laplacian

For the solution of $$ \begin{cases} \lambda u^\epsilon - \frac{\epsilon^2}{2} \Delta u^\epsilon = 0 &\text{in } \Omega \\ u^\epsilon=1 & \text{on } \partial \Omega \end{cases} $$ Varadhan ...
Jun's user avatar
  • 303
1 vote
0 answers
526 views

Martingales associated with heat equation

I am trying to learn the connection between Brownian motion and heat equation (in the spirit of Feynman-Kac, for example, here). I read (Michael E. Taylor's PDE book, Volume II, Chapter 11, ...
SMS's user avatar
  • 1,407
2 votes
0 answers
52 views

Reference Request: Dirichlet operators with singular coefficients

Let $d\geq 2$, $\delta \in (0,1)$ and let $\mathcal{L}_{d,\delta}$ be the second order differential operator defined by \begin{align*} \mathcal{L}_{d,\delta}(f)(x) = \Delta(f)(x)-\delta \|x\|^{\delta-...
user69642's user avatar
  • 778
8 votes
4 answers
2k views

How to interpret couplings in optimal transport?

Let $\mu$ and $\nu$ be two measures on some (at least measurable) space $X$. In optimal transport theory, Monge's problem to $$ \text{minimize} \quad \int c(x,T(x))\mu(dx) \quad \text{over measurable ...
vaoy's user avatar
  • 309
1 vote
0 answers
102 views

Dislocations and Random Matrix Theory

Does anyone have a good reference book that works as a good starting point for an analyst to learn about the connection between Random Matrix Theory and Dislocations? Thank you for your help. By ...
spaceman's user avatar
  • 595
2 votes
0 answers
146 views

Exit time for Brownian motion with stochastic barriers

I am interested in the expected exit time of a one-dimensional Brownian particle from a stochastically evolving interval as follows. Context: If $L_t$ and $R_t$ denote the distance to the left and ...
as1's user avatar
  • 91
2 votes
2 answers
155 views

Existence of classical solution for a parabolic equation without Hölder continuity in time for its coefficients

Consider equation $$\partial_t u = \partial_x u + \partial_{xx} u - c u + f, \hbox{ on } (t, x) \in (0, \infty) \times \mathbb R$$ with initial condition $u(0, x) = g(x).$ Suppose that $c(t, x)$ and $...
kenneth's user avatar
  • 1,399