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Henry.L
Henry.L
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To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO.

To make @Mateusz's idea more precise, what we said is that Wiener process $W_0$ can be decomposed into summation of random walks and $${\displaystyle \limsup _{n\to \infty }{\frac {W_{n}}{\sqrt {2n\log \log n}}}=1,\quad {\text{a. s.}},}$$

and the resulting $\sqrt {2n\log \log n}$ is of Hölder modulus $1/2$. Though this is a strong evidence, it is not enough since what you were asking is essentially whether the sample paths are all Hölder at $1/2$.

It is worth pointing out that Wiener processes have the self-similarity property $$\forall\lambda > 0, W_t\overset{d}{=} \frac{1}{\sqrt{\lambda }}W_{\lambda t}$$, which fails to hold for any other power of $\lambda^\alpha$ for $\alpha<-1/2$. This gives a nice image about the Hölder continuity of resulting sample path because we can always regard the sample path as sort of iterated contractions.

Following this route, [PH] Corollary 2.14 assertedindicated that the answer to your question is NO. It even fails to be continuous locally. In case of exactly $1/2$, I will say it is not based on the constructive proof in [FP]. If you want to know more about smoothness of sample paths, I wrote another answer [MO].

 

A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.

 

[PH]Peter Hansen, BROWNIAN MOTION AND HAUSDORFF DIMENSION http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf

[MO]Is there a differentiable random walk?

[FP]Friz, Peter K., and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010.

To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO.

To make @Mateusz's idea more precise, what we said is that Wiener process $W_0$ can be decomposed into summation of random walks and $${\displaystyle \limsup _{n\to \infty }{\frac {W_{n}}{\sqrt {2n\log \log n}}}=1,\quad {\text{a. s.}},}$$

and the resulting $\sqrt {2n\log \log n}$ is of Hölder modulus $1/2$. Though this is a strong evidence, it is not enough since what you were asking is essentially whether the sample paths are all Hölder at $1/2$.

It is worth pointing out that Wiener processes have the self-similarity property $$\forall\lambda > 0, W_t\overset{d}{=} \frac{1}{\sqrt{\lambda }}W_{\lambda t}$$, which fails to hold for any other power of $\lambda^\alpha$ for $\alpha<-1/2$. This gives a nice image about the Hölder continuity of resulting sample path because we can always regard the sample path as sort of iterated contractions.

Following this route, [PH] Corollary 2.14 asserted that the answer to your question is NO. It even fails to be continuous locally. In case of exactly $1/2$, I will say it is not based on the constructive proof in [FP]. If you want to know more about smoothness of sample paths, I wrote another answer [MO].

A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.

[PH]Peter Hansen, BROWNIAN MOTION AND HAUSDORFF DIMENSION http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf

[MO]Is there a differentiable random walk?

[FP]Friz, Peter K., and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010.

To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO.

To make @Mateusz's idea more precise, what we said is that Wiener process $W_0$ can be decomposed into summation of random walks and $${\displaystyle \limsup _{n\to \infty }{\frac {W_{n}}{\sqrt {2n\log \log n}}}=1,\quad {\text{a. s.}},}$$

and the resulting $\sqrt {2n\log \log n}$ is of Hölder modulus $1/2$. Though this is a strong evidence, it is not enough since what you were asking is essentially whether the sample paths are all Hölder at $1/2$.

It is worth pointing out that Wiener processes have the self-similarity property $$\forall\lambda > 0, W_t\overset{d}{=} \frac{1}{\sqrt{\lambda }}W_{\lambda t}$$, which fails to hold for any other power of $\lambda^\alpha$ for $\alpha<-1/2$. This gives a nice image about the Hölder continuity of resulting sample path because we can always regard the sample path as sort of iterated contractions.

Following this route, [PH] Corollary 2.14 indicated that the answer to your question is NO. It even fails to be continuous locally. In case of exactly $1/2$, I will say it is not based on the constructive proof in [FP]. If you want to know more about smoothness of sample paths, I wrote another answer [MO].

 

A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.

 

[PH]Peter Hansen, BROWNIAN MOTION AND HAUSDORFF DIMENSION http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf

[MO]Is there a differentiable random walk?

[FP]Friz, Peter K., and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010.

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Henry.L
Henry.L
  • 8.1k
  • 8
  • 48
  • 74

To take a different view of the question, what you are asking is whether the sample paths of this Wiener process $W_0$ can be "more smooth" in terms of Hölder continuity. The answer is NO.

To make @Mateusz's idea more precise, what we said is that Wiener process $W_0$ can be decomposed into summation of random walks and $${\displaystyle \limsup _{n\to \infty }{\frac {W_{n}}{\sqrt {2n\log \log n}}}=1,\quad {\text{a. s.}},}$$

and the resulting $\sqrt {2n\log \log n}$ is of Hölder modulus $1/2$. Though this is a strong evidence, it is not enough since what you were asking is essentially whether the sample paths are all Hölder at $1/2$.

It is worth pointing out that Wiener processes have the self-similarity property $$\forall\lambda > 0, W_t\overset{d}{=} \frac{1}{\sqrt{\lambda }}W_{\lambda t}$$, which fails to hold for any other power of $\lambda^\alpha$ for $\alpha<-1/2$. This gives a nice image about the Hölder continuity of resulting sample path because we can always regard the sample path as sort of iterated contractions.

Following this route, [PH] Corollary 2.14 asserted that the answer to your question is NO. It even fails to be continuous locally. In case of exactly $1/2$, I will say it is not based on the constructive proof in [FP]. If you want to know more about smoothness of sample paths, I wrote another answer [MO].

A side commment is that "measure concentration" seems to have slightly different meanings across communities of analysts and probabilists. In terms of analysis, people usually expect certain measures being supported on a certain set, as your example indicated, $W_0$ supported on curves of certain Hölder continuous smooth curves. In terms of probability, another (slightly different) meaning of "measure concentration" refers to the tail-behavior of the (probability, or at least bounded) measure under consideration. For example, Hoeffding inequality tells you about that there is a high probability that the probability measure "concentrates" around its mean. I guess this is worth of clarifying, at the very least.

[PH]Peter Hansen, BROWNIAN MOTION AND HAUSDORFF DIMENSION http://www.math.uchicago.edu/~may/VIGRE/VIGRE2011/REUPapers/Hansen.pdf

[MO]Is there a differentiable random walk?

[FP]Friz, Peter K., and Nicolas B. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010.