User Akira

10 votes

0 answers

422 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

737 views

How is the Gronwall lemma used in this paper?

asked Feb 23, 2023 at 11:09

6 votes

1 answer

203 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

456 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

403 views

When are the transition densities of an SDE symmetric?

asked Sep 30, 2023 at 6:46

4 votes

1 answer

773 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

2 answers

256 views

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

asked Nov 2, 2023 at 12:27

4 votes

2 answers

485 views

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

asked Nov 8, 2023 at 23:05

4 votes

1 answer

249 views

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

asked Jan 1 at 16:42

4 votes

1 answer

492 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

4 votes

0 answers

233 views

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

asked Sep 1, 2023 at 9:27

4 votes

2 answers

252 views

Hausdorff dimension of the non-differentiability set a convex function

asked Jun 30, 2022 at 19:20

4 votes

1 answer

668 views

Optimal Transport: how is this transport map Borel measurable?

asked Dec 20, 2022 at 0:21

4 votes

0 answers

92 views

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

asked Dec 3 at 9:28

3 votes

0 answers

55 views

Unique weak solution of an SDE for a general initial distribution

asked Jul 5 at 8:32

3 votes

1 answer

263 views

Hölder continuity in time of heat semigroup

asked Jul 1 at 11:45

3 votes

2 answers

937 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

132 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

145 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

2 votes

0 answers

111 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

154 views

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

asked Feb 26 at 9:00

2 votes

1 answer

272 views

Decompose a function into a bounded part and a Lipschitz part

asked Feb 27 at 11:42

2 votes

1 answer

216 views

Decay estimate of moment of an SDE

asked Mar 3 at 15:32

2 votes

2 answers

191 views

Gronwall's inequality in discretized time

asked May 13 at 20:14

2 votes

1 answer

246 views

Does $X_t$ with $t>0$ admit a density?

asked May 14 at 8:32

2 votes

0 answers

66 views

Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?

asked May 14 at 20:20

2 votes

1 answer

154 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

1 answer

65 views

Approximate a non-negative function which is measurable in product $\sigma$-algebra

asked May 23 at 7:47

2 votes

1 answer

392 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

79 views

Does this variant coincide with the usual Hölder space?

asked Jun 28, 2023 at 8:37

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

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?

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

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

Hausdorff dimension of the non-differentiability set a convex function

Optimal Transport: how is this transport map Borel measurable?

Hölder stability of the PDE $\partial_t u (t, x) = \operatorname{div} \{ a (t, x) \nabla u(t, x) \}$

Unique weak solution of an SDE for a general initial distribution

Hölder continuity in time of heat semigroup

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?

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

Gronwall's inequality in discretized time

Does $X_t$ with $t>0$ admit a density?

Is $F: \mathbb T \times \mathbb R^d \times \Omega \to \mathbb R^d$ (constructed from Itô integral) Borel measurable in the product $\sigma$-algebra?

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

Approximate a non-negative function which is measurable in product $\sigma$-algebra

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?