User Akira

10 votes

0 answers

395 views

Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

asked Jan 19 at 6:22

7 votes

1 answer

640 views

How is the Gronwall lemma used in this paper?

asked Feb 23, 2023 at 11:09

6 votes

1 answer

182 views

Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?

asked Dec 29, 2022 at 12:54

5 votes

1 answer

382 views

Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?

asked Sep 3, 2023 at 21:41

4 votes

1 answer

355 views

When are the transition densities of an SDE symmetric?

asked Sep 30, 2023 at 6:46

4 votes

1 answer

743 views

How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?

asked Oct 7, 2023 at 8:37

4 votes

0 answers

173 views

References for derivative w.r.t. initial condition of an ODE

asked Sep 1, 2023 at 9:27

4 votes

2 answers

193 views

Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?

asked Nov 2, 2023 at 12:27

4 votes

2 answers

464 views

How to get this inequality in Santambrogio's book about optimal transport?

asked Nov 8, 2023 at 23:05

4 votes

1 answer

188 views

Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?

asked Jan 1 at 16:42

4 votes

1 answer

531 views

Optimal Transport: how is this transport map Borel measurable?

asked Dec 20, 2022 at 0:21

3 votes

2 answers

227 views

Hausdorff dimension of the non-differentiability set a convex function

asked Jun 30, 2022 at 19:20

3 votes

2 answers

795 views

Does this version of Clairaut-Schwarz theorem hold when mixed partial derivatives are of order greater than $2$?

asked Nov 16, 2019 at 8:12

3 votes

1 answer

126 views

If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry

asked Mar 4, 2023 at 17:53

3 votes

1 answer

107 views

Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?

asked Oct 23, 2023 at 16:59

3 votes

1 answer

353 views

Does $f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$ imply $f=0$?

asked Mar 26 at 22:32

2 votes

1 answer

120 views

Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

asked Mar 26 at 16:40

2 votes

0 answers

104 views

Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

asked Feb 15 at 6:18

2 votes

1 answer

146 views

Are these two norms on localized versions of $L^p_q$ equivalent?

asked Feb 26 at 9:00

2 votes

1 answer

246 views

Decompose a function into a bounded part and a Lipschitz part

asked Feb 27 at 11:42

2 votes

1 answer

193 views

Decay estimate of moment of an SDE

asked Mar 3 at 15:32

2 votes

1 answer

333 views

Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?

asked Mar 21, 2023 at 16:56

2 votes

0 answers

76 views

Does this variant coincide with the usual Hölder space?

asked Jun 28, 2023 at 8:37

2 votes

1 answer

165 views

Is the Lipschitz constant of $f$ equal to $\|\nabla f\|_{L^\infty}$?

asked Jul 25, 2023 at 15:45

2 votes

1 answer

229 views

Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

asked Mar 4, 2023 at 4:06

2 votes

1 answer

131 views

Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?

asked Mar 4, 2023 at 9:19

2 votes

0 answers

175 views

Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?

asked Nov 20, 2022 at 10:57

2 votes

0 answers

83 views

A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged

asked Nov 21, 2022 at 12:45

1 vote

1 answer

196 views

Optimal transport: the existence of an optimal pair of $c$-conjugate functions

asked Nov 21, 2022 at 13:57

1 vote

1 answer

102 views

Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?

asked Nov 28, 2022 at 19:02

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74 Questions

Upper bound Hölder norm of the solution to the non-linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (u (t, x))|^2 u(t, x) \}$

How is the Gronwall lemma used in this paper?

Can Theorem 1.40 in Rudin's Real and Complex Analysis be strengthened when the $\sigma$-algebra is Borel?

Is there a Borel measurable $f:\mathbb{R}^d \to \mathbb{R}^d$ such that $f(x) \in \partial \varphi (x)$ for all $x$?

When are the transition densities of an SDE symmetric?

How to get $\int_{\mathbb R^d} |\partial_i\partial_j(1-\Delta)^{-\frac{\delta}{2}}p_t(\cdot-y)(x)| \, \mathrm d x \lesssim t^{\frac{\delta}{2}-1}$?

References for derivative w.r.t. initial condition of an ODE

Are the sublevel sets of Boltzmann entropy compact in Wasserstein distance?

How to get this inequality in Santambrogio's book about optimal transport?

Weak uniqueness of an SDE with locally Lipschitz drift and additive noise?

Optimal Transport: how is this transport map Borel measurable?

Hausdorff dimension of the non-differentiability set a convex function

Does this version of Clairaut-Schwarz theorem hold when mixed partial derivatives are of order greater than $2$?

If $X,X'$ have the same $\varepsilon$-packing numbers and $f:X \to X'$ surjective $1$-Lipschitz, then $f$ is an isometry

Let $\mu : [0, T] \to \mathcal P_2^a (\mathbb R^d), t \mapsto \mu_t$ be absolutely continuous. Is $t \mapsto \mathcal H (\mu_t)$ continuous?

Does $f(t) \le \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$ imply $f=0$?

Grönwall-type inequality for $f(t) \le \alpha + \int_0^t (t-s)^{-\frac{1}{2}} [f(s) + |f(s)|^{\beta}] \, \mathrm d s$

Upper bound Hölder norm of the solution to the linear PDE $\partial_t u (t, x) = \Delta_x \{ |\sigma (x)|^2 u(t, x) \}$

Are these two norms on localized versions of $L^p_q$ equivalent?

Decompose a function into a bounded part and a Lipschitz part

Decay estimate of moment of an SDE

Interacting particle system: how are the particles independent conditionally to the knowledge of their initial positions?

Does this variant coincide with the usual Hölder space?

Is the Lipschitz constant of $f$ equal to $\|\nabla f\|_{L^\infty}$?

Are two metric spaces isometric if they have the same $\varepsilon$-covering numbers for all $\varepsilon>0$?

Are two metric spaces isometric if they have the same $\varepsilon$-covering and $\varepsilon$-packing numbers for all $\varepsilon>0$?

Optimal transport: how is the use of disintegration theorem valid in this construction of $\widetilde{\phi}$?

A variant of disintegration theorem where the assumptions on $f$ and $g$ are exchanged

Optimal transport: the existence of an optimal pair of $c$-conjugate functions

Does the topology of Wasserstein space $(\mathcal P_p (E), W_p)$ coincide with the initial topology induced by $\mathcal C_b(E) \cup \{g_p\}$?