# I Know You Did Not Write That! A Sampling Based Watermarking Method for
Identifying Machine Generated Text

Kaan Efe Keleş
TOBB ETU
<EMAIL_ADDRESS>
&Ömer Kaan Gürbüz
Bilkent University
<EMAIL_ADDRESS>
&Mucahid Kutlu
TOBB ETU
<EMAIL_ADDRESS>

###### Abstract

Potential harms of Large Language Models such as mass misinformation and
plagiarism can be partially mitigated if there exists a reliable way to detect
machine generated text. In this paper, we propose a new watermarking method to
detect machine-generated texts. Our method embeds a unique pattern within the
generated text, ensuring that while the content remains coherent and natural
to human readers, it carries distinct markers that can be identified
algorithmically. Specifically, we intervene with the token sampling process in
a way which enables us to trace back our token choices during the detection
phase. We show how watermarking affects textual quality and compare our
proposed method with a state-of-the-art watermarking method in terms of
robustness and detectability. Through extensive experiments, we demonstrate
the effectiveness of our watermarking scheme in distinguishing between
watermarked and non-watermarked text, achieving high detection rates while
maintaining textual quality.

##  1 Introduction

Transformer based Large Language Models (LLMs) (Vaswani et al., 2023) such as
ChatGPT, Llama2 (Touvron et al., 2023) are able to generate texts that closely
resemble human authored texts. For instance, Clark et al. (2021) report that
untrained humans are not able to distinguish between texts generated by GPT-3
and texts authored by humans. As we train larger models with more parameters
on an ever-expanding corpora, their capabilities in generating human-like text
are likely to increase (Hoffmann et al., 2022). With their incredible
performance in text generation, they become effective tools for automating
monotonous text based tasks such as summarization and translation (Radford et
al., 2019).

However, these LLMs pose various threats to society because they can be also
used for bad causes such as generating credible-sounding misinformation (Pan
et al., 2023), creating fake product reviews (Adelani et al., 2019) and
academic plagiarism (Dehouche, 2021). Recent studies have discovered that even
though LLM-generated responses may sound convincing, they can be frequently
incorrect (Lin et al., 2022).

The potential negative consequences associated with LLMs can be reduced
significantly if a reliable detection system is in place to differentiate
between machine-generated and human-written texts. A number of researchers
focused on this important problem and proposed various approaches training a
classifier (Guo et al., 2023), detecting based on linguistic features (Guo et
al., 2023) and log probabilities and perturbations (Mitchell et al., 2023).
Data driven methods such as training classifiers requires wide range of data
with different styles, sources, and languages. More importantly non-
watermarking detectors are generally found to be ineffective (Krishna et al.,
2023). Moreover, existing perplexity based detectors are biased against non-
native English writers (Liang et al., 2023), raising ethical concerns about
their usage in real-life applications.

In this paper we propose a novel model-agnostic watermarking method to detect
machine generated text. In watermarking, a hidden pattern is inserted to a
passage that is imperceptible to humans but can be easily detected an
algorithm.

In our proposal, we interfere with the randomness of sampling a new token to
be generated in the decoding phase of LLMs. For each token to be generated, we
sample multiple candidate tokens based on their probability provided by the
LLM and calculate a secret number for each of the candidate tokens.
Subsequently, we pick the token with the highest secret number value. The way
we calculate the secret number enables us to retrieve the same values from
generated text. And our maximization effort lets us discriminate against non-
watermarked text.

In our experiments, we evaluate the quality of the watermarked texts and how
accurately we can detect the watermarks using various datasets and LLMs. We
also compare our model against watermarking method of Kirchenbauer et al.
(2023a). In our experiments, we show that we are able to detect watermarked
texts almost in all cases. In addition, we observe that our method based on
sampling with replacement does not reduce the text quality in almost all cases
while our method based on sampling without replacement yields slight decrease
in text quality. In addition, we show that our proposed method is robust to
token level paraphrasing attacks.

The main contributions of our work are as follows. i) We introduce a novel
watermarking scheme to detect machine-generated text. In our comprehensive
evaluation we show that our watermarks are highly detectable while causing a
slight decrease in text quality. ii) We share both our code and dataset to
ensure reproducibility of our results and help other researchers build upon
our findings.111The code will be made available soon..

##  2 Related Work

The remarkable achievements of Large Language Models (LLMs) compelled
researchers to shift their attention towards understanding their potential
drawbacks and risks. We direct readers to the survey studies conducted by
Crothers et al. (2022) and Weidinger et al. (2021) for an in-depth analysis of
the risks associated with LLMs. Now, we focus on studies on detecting texts
generated by LLMs.

###  2.1 Non-Watermarking Detection Methods

Gehrmann et al. (2019) propose a tool GLTR which works in a white-box setting
and highlights texts based on probability distribution of tokens provided by
the LLMs. They show that their visual tool improves the human detection rate
of machine generated text from 54% to 72% without any prior training and
without tampering with the text generation phase.

Mitchell et al. (2023) also work in a white-box setting and create
perturbations of the candidate text and analyze the negative curvature regions
of the model’s log probability function. Their main hypothesis for detection
is as follows. When machine generated text is modified it tends to have lower
log probability. However, modifications on the human-written text may have
higher or lower log probability than the unmodified text.

Zellers et al. (2019) examine several schemes to detect fake news article
using GROVER which is a language model that generates and classifies fake news
articles. They conclude that the most effective model for identifying fake
news generated by GROVER is the model itself. Adelani et al. (2019) also
report that GROVER is highly accurate in detecting fake reviews. Zellers et
al. (2019) argue that machine-generated text classification requires a similar
inductive bias as the generator model, rather than expressive capability.
However, these findings differ from those of Solaiman et al. (2019) as they
claim that a fine-tuned RoBERTa model is a more effective detector than a
similarly-capable fine-tuned GPT-2 model.

A number of researchers focused on developing machine learning models to
identify generated texts. For instance, Fagni et al. (2021) report that
transformer based classifiers to be the best discriminators of fake tweets.
Guo et al. (2023) collect a dataset of ChatGPT’s and experts’ responses to
questions in various topics such as finance and medicine. and train
classifiers to predict if a given passage is computer generated. A similar
approach is also followed by the creators of ChatGPT with underwhelming
results222https://openai.com/blog/new-ai-classifier-for-indicating-ai-written-
text/. In our work, we propose a watermarking method to detect generated
texts. We discuss watermarking methods in the literature in the following
section.

###  2.2 Watermarking Detection Methods

Abdelnabi and Fritz (2020) introduce Adversarial Watermarking Transformer
(AWT) model, which encodes binary messages in text to trace its origin and
prevent malicious use, using a jointly trained encoder-decoder and adversarial
training, ensuring the watermark is discreet while maintaining the text’s
original meaning. Ueoka et al. (2021) proposes using a masked language model,
which has a high payload capacity and is less susceptible to automatic
detection than generation-based methods.

The closest work to our own is Kirchenbauer et al. (2023a)’s watermarking
method. They propose selecting a randomized subset of approved tokens from the
vocabulary and then promoting the sampling of the tokens from chosen approved
subset of the vocabulary via increasing the subsets logits. The randomization
is seeded on previously generated token(s) in a context window. In our work,
we interfere the sampling process without changing LLMs’ probability
distribution over vocabulary while Kirchenbauer et al. (2023a) interfere the
probability distribution. In our experiments, we extensively compare our
proposed method against Kirchenbauer et al. (2023a)’s.

Watermarking methods are investigated to protect text generation models
against model extraction attacks, Zhao et al. (2023) inject a sinusoidal
signal into the model’s generation probabilities. As a result, the suspect
model is anticipated to retain the watermark from the victim model, rendering
it detectable.

###  2.3 Paraphrasing Attacks

As there are tools to detect generated texts, people might want to avoid these
detection tools by intentionally changing the generated texts. Therefore,
prior work also explored how vulnerable detection systems are against
paraphrasing attacks.

Sadasivan et al. (2023) demonstrate how effective off-the-shelf sentence-level
paraphrasing models can be at evading detection and conclude that detecting
generated text is an unsolvable problem. However, this conclusion is
contradicted by Chakraborty et al. (2023) as they show that detection should
always be possible when there exist enough samples. Krishna et al. (2023)
develop a paraphrasing model which successfully evades several detectors
including watermarking (Kirchenbauer et al., 2023a) and DetectGPT (Mitchell et
al., 2023). They conclude that non-watermarking detectors to be generally
ineffective. In their proposed detection scheme, the API provider maintains a
database containing every sequence generated by their LLM. When a detection
query is initiated, this database is queried to identify a previously-
generated sequence that exhibits the highest semantic similarity to the query.
If the level of similarity surpasses a predefined threshold, the query is
classified as machine-generated.

##  3 Problem Definition

Our goal is to develop a model-agnostic watermarking method to identify
generated texts. Let $LLM$ be a large language model and $LLM^{w}$ is its
version with watermarking feature. In addition, let
$T_{LLM}(P)$/$T_{LLM}^{w}(P)$ be a text generated by $LLM$/$LLM^{w}$ for the
given prompt $P$. An ideal watermarking method should have the following
properties:

  * •

The watermarking process should not decrease the quality of the texts, i.e.,
the quality of $T_{LLM}(P)$ and $T_{LLM}^{w}(P)$ should be similar for any
given $P$.

  * •

Watermarking text should not necessitate retraining or fine-tuning.

  * •

We should have the capability to compute a statistical confidence interval
with interpretable values for the detection and sensitivity analysis of the
watermark.

  * •

The watermark should be robust to perturbations. An adversary must make
significant modifications to remove the watermark.

##  4 Proposed Methodology

In this section, we explain our proposed method to generate watermarked text
(Section 4.1) and how to detect the watermark within a given text (Section
4.2).

###  4.1 Generating Watermarked Texts

In our watermarking method, we interfere with the randomness of picking the
next token according to its conditional probability provided by a language
model in the decoding stage. The details of our method are shown in Algorithm
1.

Algorithm 1 Text Generation with the Sampling Watermarker

1:Input: P {Prompt given to the model}

2:Parameter-1: $y$ {The sampling count}

3:Parameter-2: $k$ {The context window size}

4:$T_{LLM}(P)=P$ {Keeps the whole text}

5:for each token to be generated do

6: $D=LLM(T_{LLM}(P)))$ {Get the probability distribution from the LLM}

7: $C_{[1-y]}$ = $sample(D,y)$ {Sample y candidate tokens}

8: for $i\in\\{1,\dots,y\\}$ do

9: $S^{C_{i}}=RNG(seed=hash(T_{LLM}(P)^{[N,N-k]},C_{i}))$ {Calculate the
secret number}

10: end for

11: $T_{LLM}(P)=T_{LLM}(P)+C^{argmax(S^{C_{1}},\cdots,S^{C_{y}})}$
{Concatenate the selected token}

12:end for

For a given input prompt $P$, $LLM$ produces a text T in an iterative way
[Lines 1-9]. In each iteration, LLM outputs a conditional probability
distribution vector over the vocabulary $V$ for the next token to be generated
[Line 3]. We multinomially sample $y$ candidate tokens based on the
probability distribution vector [Line 4]. Subsequently, we compute a secret
number for each candidate token $t$ [Lines 5-7]. In order to compute the
secret number of a candidate token ($S^{t}$), we first concatenate the $k$
previous tokens and the candidate token $t$ and then calculate their SHA256
hash value. Subsequently, we seed a random number generator with the hash
value [Line 6] and generate a random number. Next we pick the token with the
highest secret number for the next token [Line 8].

The secret number of any token in a candidate passage only depends on itself
and the $k$ tokens that precede it. This enables us to retrieve the same
secret number for every token in a passage outside of the generation process.
Moreover, if a passage is watermarked we expect the average secret number of
the tokens that make up the text to be significantly higher than otherwise.
This is because while the production of the non-watermarked text is completely
ignorant of the secret numbers of tokens, our watermarking scheme actively
attempts to maximize this value.

During sampling, we have the option to sample candidate tokens with or without
replacement. When we sample without replacement, the secret numbers of the
candidate tokens are guaranteed to be distinct values. Maximizing the use of
distinct values tends to result in larger secret number values, making the
watermark more detectable. On the other hand, if the entropy of the
probability distribution is low, i.e., there are few plausible tokens to be
generated, sampling without replacement would cause the model to pick the
unlikely tokens, reducing the quality of the generated text. Therefore, we
also explore sampling with replacement and evaluate the impact of both
sampling methods in Section 5.

###  4.2 Detecting the watermark

In order to detect whether a given text X is watermarked or not, i.e., a text
generated by our scheme or not, we first tokenize X and calculate the secret
number of each token in X. The secret number of the $r^{th}$ token of X can be
calculated as follows.

$S^{X_{r}}=RNG(seed=hash(X_{(r-k)}\cdots X_{(r-1)},X_{(r)}))$ (1)

where $RNG$ is a random number generator which draws values from a continuous
uniform distribution spanning the interval from zero to one. The anticipated
mean of the secret number for the tokens composing a text aligns with a normal
distribution characterized by an expected mean of 0.5 and an expected variance
of $\frac{1}{12*N}$ (See Blitzstein and Hwang (2015) for explanation), where
$N$ represents the number of tokens within the given text X. As the length of
the candidate text increases, the average secret number for non-watermarked
text gradually approaches this theoretical distribution with diminishing
variance, thus reducing the likelihood of the text’s average secret number
deviating significantly from 0.5. Conversely, during the watermarking process,
tokens are selected from a set of candidates based on their possession of the
highest secret number (out of $y$ candidates). This selection dramatically
alters the distribution of the average secret number, rendering it exceedingly
improbable for the text to have arisen through natural generation. Thus, we
classify the text as watermarked if a certain threshold is exceeded. Formally,
we define the following null hypothesis.

$H_{0}$: The text sequence is generated without any attempt to maximize the
secret number average.

The formula of the z-score for testing the hypothesis is as follows:

$z-score=(\overline{sna}-0.5)/\sqrt{1/(12\cdot N)}$ (2)

where $\overline{sna}$ denotes the secret number average of the candidate text
and $N$ represents how many tokens make up the candidate text. The null
hypothesis is rejected (and the watermark is detected) if $z-score$ is above a
chosen threshold $u$.

##  5 Experiments

###  5.1 Experimental Setup

In this section, we explain evaluation metrics (Section 5.1.1) to assess the
quality of our watermarking method, describe the models we used for
watermarking (Section 5.1.2), baseline methods we compare against our methods
(Section 5.1.3), and datasets we utilized in our experiment (Section 5.1.4).
Lastly, we provide details about implementation details (Section 5.1.5).

####  5.1.1 Evaluation Metrics

In order to measure the quality of watermarking methods, we focus on the
quality of the generated text and our detection rate. We adopt the measures
used by related prior work (Kirchenbauer et al., 2023b; Krishna et al., 2023).
In particular, we calculate how the generated texts are similar to the human
authored ones using P-SP (Wieting et al., 2023). In addition, we use diversity
which aggregates n-gram repetition rates. A high diversity score represents a
more diverse text where fewer n-grams are repeated (Li et al., 2023). Given
the fraction of unique $n$-grams (which is denoted as $u_{n}$) diversity up to
the $N^{th}$ order is defined as follows.

$\textnormal{diversity}=-\log{\left(1-\prod_{n=1}^{N}(1-u_{n})\right)}$ (3)

Lastly, we use coherence to measure the semantic coherence between the prompt
and the generated text. We employ the sentence embedding method, SimCSE (Gao
et al., 2022) for this calculation. Given the prompt $\boldsymbol{x}$ and the
generated text $\hat{\boldsymbol{x}}$, the coherence score is defined as
$v_{\boldsymbol{x}}^{\top}v_{\hat{\boldsymbol{x}}}/(\|v_{\boldsymbol{x}}\|\cdot\|v_{\hat{\boldsymbol{x}}}\|)$,
where $v_{\boldsymbol{x}}=\textup{SimCSE}(\boldsymbol{x})$ and
$v_{\hat{\boldsymbol{x}}}=\textup{SimCSE}(\hat{\boldsymbol{x}})$.

####  5.1.2 Models

As our approach can be applied in any model, we utilize three different models
that our hardware systems could execute. In particular, we use OPT (Zhang et
al., 2022) with 1.3B parameters, BTLM-3B (Dey et al., 2023) with 3B
parameters, and Llama2 (Touvron et al., 2023) with 7B parameters.

####  5.1.3 Baseline Methods

We compare our proposed method against Kirchenbauer et al. (2023a)’s study,
which is also known as “Maryland Watermark” (MWM). For the configuration
parameters of their approach, we set the greenlist fraction $\gamma$ at 0.25
and set the logit bias $\delta$ to 2.

####  5.1.4 Datasets

In our experiment, we use two different datasets: i) the train split of the
’realnewslike’ portion of the C4 (stands for “Colossal Clean Crawled Corpus”)
dataset (Raffel et al., 2020) and ii) the train split for Wikitext
(103-v1-raw) dataset (Merity et al., 2016). C4 is an extensive web text
collection resembling real news articles while Wikitext consists of 100M
tokens extracted from the set of verified Good and Featured articles on
Wikipedia, providing a more structured and manageable source.

We use the first 100 tokens of the passages as prompts. In order to have a
fair comparison, we use 200 tokens for all cases. Therefore, we allow models
to generate maximum 200 new tokens. For a given prompt, if any of the
generated text is less than 200 tokens, we discard it, and try another prompt
drawn from the corresponding dataset. We continue this process until we reach
500 samples for each dataset. Eventually, for each dataset and model we use,
we create five text subdatasets: i) texts generated by Maryland watermarking
($T_{MWM}$), ii) texts generated by our approach with sampling with
replacement ($T_{SWR}$), iii) texts generated by our approach with sampling
without replacement ($T_{SWOR}$), iv) texts generated without watermark
($T_{NoWM}$), and v) texts authored by humans ($T_{Humans}$).

####  5.1.5 Implementation

We implemented the sampling watermarker using the PyTorch (Paszke et al.,
2019) backend of the Hugging Face library (Wolf et al., 2019). We utilized the
generate API provided by Hugging Face for generating text. This API allows for
passing a custom LogitsProcessor which can be used to modify the prediction
scores of a language model head for generation. We use Top-k sampling (Fan et
al., 2018) with $top-k=40$ before doing any sampling on all methods. For our
proposed method we set the context window size $k$ to 1 and sampling count $y$
to 5 unless otherwise is mentioned.

###  5.2 Experimental Results

This section comprises of four subsections, each serving distinct research
objectives. The first (Section 5.2.1) assesses watermark detectability, the
second (Section 5.2.2) examines textual quality under watermarking, the third
(Section 5.2.3) evaluates watermark robustness against attacks, and the final
subsection (Section 5.2.4) investigates the impact of various generation
parameters on watermarking performance.

####  5.2.1 Detectibility Experiments

In this experiment, we assess how accurate watermark detection mechanisms
work. Specifically, we run our watermarking methods and MWM for all datasets
we create and calculate average z-scores over the generations. In addition, we
set the z-score threshold ($u$) to 4 for both watermarking schemes as in
(Kirchenbauer et al., 2023a) and calculate the percentage of the texts
detected as watermarked. The results are shown in Table 1.

|  | C4 | Wikitext
---|---|---|---
|  | OPT-1.3B | BTLM-3B | Llama2-7B | OPT-1.3B | BTLM-3B | Llama2-7B
Text | Detector | z-score | %WM | z-score | %WM | z-score | %WM | z-score | %WM | z-score | %WM | z-score | %WM
$T_{SWR}$ | SWR | 11.31 | 99.8% | 10.11 | 99.8% | 9.44 | 99% | 12.09 | 99.8% | 10.33 | 100% | 10.36 | 99.8%
$T_{SWOR}$ | SWOR | 16.85 | 100% | 16.29 | 100% | 16.66 | 100% | 16.92 | 100% | 16.26 | 100% | 17.23 | 100%
$T_{MWM}$ | MWM | 10.77 | 100% | 9.82 | 100% | 9.71 | 99.4% | 11.79 | 100% | 10.43 | 100% | 10.65 | 97%
$T_{Humans}$ | SWR | 0.27 | 0% | -0.07 | 0% | 0.22 | 0% | 0.03 | 0% | -0.05 | 0% | 0.28 | 0%
MWM | -0.23 | 0% | -0.46 | 0.2% | 0.21 | 0.2% | 0.35 | 0.6% | 0.21 | 0.2% | -0.01 | 0.2%
$T_{NoWM}$. | SWR | 0.22 | 0% | -0.25 | 0% | 0.44 | 1.4% | 0.69 | 0.6% | -0.22 | 0% | 0.17 | 3.6%
MWM | -0.25 | 0% | -0.42 | 0.2% | 0.32 | 1% | 0.01 | 0.4% | -0.17 | 0.2% | 0.39 | 3.4%

Table 1: The average z-scores over the generations when attempted to detect
the watermark and the ratio of samples detected as “watermarked” by the
corresponding detector. The text in bold represent the highest z-score for
watermarked text and lowest for baseline completion text.

The average z-scores exceed 10 in most of the watermarked texts, and is near 0
for non-watermarked text, showing the effectiveness of watermarking schemes.
SWOR achieves achieves the highest z-score and detection rates in watermarked
texts.

Our watermarking methods consistently avoid false positives when applied to
human authored text, whereas MWM occasionally misidentifies such content as
watermarked. Moreover, both MWM and our approach have higher false positive
when dealing with non-watermarked machine-generated text compared to human
authored text. This is because non-watermarked machine-generated text
inherently resembles watermarked machine-generated text.

####  5.2.2 Textual Quality Experiments

In this experiment, we assess how watermarking affects the textual quality. We
report P-SP, diversity, and coherence scores in in Table 2 for texts
watermarked with our approaches, Maryland Watermarking, and without any
watermark. Additionally, for a more comprehensive understanding, we provide
samples of watermarked texts in the Appendix A.

Regarding similarity with respect to human authored text (P-SP), we observe
that MWM achieves higher scores than our methods for OPT-1.3B and BTLM-3B.
However, SWR outperforms others when Llama2-7B is used for generation.
Interestingly, SWR even yields higher P-SP score than non-watermarked text
with Llama2-7B in Wikitext. We observe a similar pattern in other metrics such
that MWM yields higher score with OPT-1.3B and BTLM-3B models than our models
in most of the cases. On the other hand, SWR outperforms others with the
largest model we use. Regarding SWOR vs. MWM with Llama2-7B is mix such that
SWOR outperform MWM in Wikitext but not in C4.

Metric |  Method | C4 | Wikitext
---|---|---|---
OPT-1.3B | BTLM-3B | Llama2-7B | OPT-1.3B | BTLM-3B | Llama2-7B
P-SP |  SWR | 0.44 | 0.48 | 0.48 | 0.45 | 0.48 | 0.52
SWOR | 0.40 | 0.42 | 0.38 | 0.42 | 0.43 | 0.44
MWM | 0.46 | 0.49 | 0.45 | 0.47 | 0.49 | 0.41
NWM | 0.47 | 0.50 | 0.48 | 0.49 | 0.49 | 0.46
Diversity |  SWR | 6.92 | 7.50 | 8.16 | 6.26 | 7.06 | 6.96
SWOR | 6.84 | 7.49 | 7.48 | 6.42 | 7.23 | 6.66
MWM | 7.40 | 7.90 | 5.88 | 6.77 | 7.46 | 5.38
NWM | 7.87 | 7.87 | 6.17 | 7.16 | 7.55 | 6.1
Coherence |  SWR | 0.63 | 0.64 | 0.64 | 0.67 | 0.66 | 0.65
SWOR | 0.58 | 0.59 | 0.53 | 0.63 | 0.60 | 0.54
MWM | 0.64 | 0.64 | 0.65 | 0.68 | 0.66 | 0.58
NWM | 0.66 | 0.66 | 0.67 | 0.70 | 0.66 | 0.62

Table 2: The impact of watermarking on the the quality of the generated text.
The highest score among watermarked texts for each case is shown in bold. MWM:
Maryland Watermarking, SWR: Sampling with replacement, SWOR: Sampling without
replacement, NWM: No Watermarking.

####  5.2.3 Robustness Experiments

In order to assess how vulnerable the watermarking methods are against token
level paraphrasing attacks, we conduct an experiment similar to the one in
(Kirchenbauer et al., 2023a). In particular, we randomly pick $\%t$ of tokens
in the watermarked and mask them. Next, we use DistilRoBERTa-Base model (Sanh
et al., 2020) to replace masked tokens, ensuring that the model did not
predict the same token that was initially masked. Figure 1 shows how different
attack percentages effect the detection of the watermarked text. Sampling
without replacement achieves high detection rates even in attacks with %40,
outperforming all other methods. Sampling with replacement and Maryland
Watermarker achieve similar detection rates.

Figure 1: The impact of paraphrasing attacks on the detection rate of
watermarked texts.

####  5.2.4 The Impact of Sampling Count

We explore the impact of the sampling count used for secret number generation,
$y$ on the quality of the generated texts and the detection rate. In
particular, we vary $y$ from 2 to 11 and generate text using our approach with
and without replacement using C4 dataset and Llama-2-7B model. Table 3 shows
the text quality metrics along with average z-score and detection rate. We
observe that increasing the sampling count $y$ results in decreasing quality
scores in all cases, but yields higher z-scores. Detection rate for SWOR
remains at %100 even at a low sampling count of $y=2$ and SWR achieves 99%
rate when $y=5$.

$y$ | P-SP | Diversity | Coherence | z-score | Detection Rate
---|---|---|---|---|---
| SWR | SWOR | SWR | SWOR | SWR | SWOR | SWR | SWOR | SWR | SWOR
2 | 0.49 | 0.45 | 8.33 | 8.65 | 0.66 | 0.61 | 4.79 | 8.33 | %76 | %100
5 | 0.48 | 0.38 | 8.16 | 7.48 | 0.64 | 0.53 | 9.44 | 16.66 | %99 | %100
8 | 0.46 | 0.34 | 7.66 | 6.4 | 0.62 | 0.50 | 11.72 | 19.51 | %100 | %100
11 | 0.45 | 0.30 | 7.65 | 5.83 | 0.62 | 0.46 | 12.91 | 20.94 | %100 | %100
Table 3: The effect of sampling count $y$ on textual quality metrics. Model:
Llama-2-7B, Dataset:c4, $k$:1.

####  5.2.5 Entropy in Probability Distribution

The effectiveness of our proposed method and the Maryland watermarking depends
on the language model’ output distribution. For instance, if the model outputs
a low entropy distribution for the next token, our sampling with replacement
based method is likely to sample the same $y$ tokens as candidates. However,
in sampling without replacement case, the watermarker is guaranteed to sample
$y$ unique tokens and pick the one that has the highest secret number.

In this experiment, we manually manipulate the output distribution entropy of
our models by adjusting the sampling temperatures to assess its impact. Table
4 shows the average z-score for varying temperature values for Llama2-7B model
on C4 dataset. As expected we observe that both SWR and MWM exhibit stronger
watermarks when the output distribution entropy is higher. SWOR shows slight
variations in the average z-score but these are just statistical noises as
SWOR is designed to be unaffected by the underlying distribution entropy.

Temperature | 0.8 | 0.9 | 1 | 1.1 | 1.2
---|---|---|---|---|---
SWR | 8.14 | 8.91 | 9.44 | 10.38 | 10.82
SWOR | 16.89 | 16.75 | 16.66 | 16.68 | 16.61
MWM | 8.02 | 8.85 | 9.71 | 10.65 | 11.24
Table 4: The effect of sampling temperature on the average z-score. Lower
temperatures yield output distributions with lower entropy vice versa. Model:
Llama2-7B, Dataset:C4, $k$:1,$y$:5

##  6 Limitations

While our work makes an important contribution in the research of LLMs, there
there are certain limitations that require further research in the future.
Firstly, we derive our prompts from two different datasets. However, the
watermarking performance highly depends on the given prompt. For instance, if
we ask a factual question to a model (e.g, what is the full text of the U.S.
constitution?), watermarking the generated output would be challenging because
of limited flexibility in the answer. Therefore, a larger number of datasets
covering diverse set of topics are required. Similarly, we ran our experiments
for three models due to hardware limitations. As the watermarking performance
is affected by the models used for text generation, covering a wider range of
LLMs is required for a more reliable assessment of the methods.

Furthermore, in our study, we focus on only the task of completing a text for
a given prompt. We acknowledge that further evaluation of the proposed
watermark across different down stream tasks such as question answering and
summarization would be beneficial. We leave this exploration as future work.

Lastly, we explore only token level paraphrasing attacks to measure the
robustness of the models. There exist different methods for manipulating text
to evade watermarking detection such as deletion, unicode attacks and human
paraphrasing. Thus, other types of attacks should be explored to further
analyze the robustness of watermarking methods.

##  7 Conclusion and Future Work

In this work, we propose a watermarking scheme which embeds a unique pattern
into the generated text while preserving its coherence and natural readability
for human readers. Specifically, We modify the token sampling process of LLMs.
In particular, we first sample multiple tokens based on probability
distribution over vocabulary and then calculate a unique secret number for
each sampled one. We always pick the token with the highest secret number,
allowing us to trace the hints of generation process.

In our experiments with multiple datasets and LLMs, we show that our method we
show that our watermarking is detectable and reduce slight decrease in text
quality. Furthermore, our method outperforms Kirchenbauer et al. (2023a)’s
method in terms of detectability and robustness. Regarding text quality, we
achieve slightly superior results compared to (Kirchenbauer et al., 2023a)
when applied to larger models, albeit with less favorable outcomes when
dealing with smaller models.

There are multiple research directions we plan to extend in the future.
Firstly, we plan to conduct our experiments on a larger scale in terms of data
and model size and types. Secondly, a more sophisticated watermark could be
implemented by adaptively choosing the sampling count $y$ based on the entropy
of the output distribution. Specifically, when the output distribution
exhibits low entropy, we can select a smaller value for $y$ and conversely,
when the entropy is high, we can opt for a larger value. This method would
ensure less perplexity on low entropy text while allowing for a stronger
watermark to be embedded on higher entropy text. We leave this extension as a
future work.

Lastly, there are no inherent obstacles to abstaining from the concurrent
application of both our and Kirchenbauer et al. (2023a)’s watermarks during
text generation. This would enable texts that are detectable by both
watermarking methods. Employing two relatively less intrusive watermarks might
potentially better maintain the textual quality while preserving high
detectability.

## References

  * Abdelnabi and Fritz (2020) Sahar Abdelnabi and Mario Fritz. 2020.  Adversarial watermarking transformer: Towards tracing text provenance with data hiding.
  * Adelani et al. (2019) David Ifeoluwa Adelani, Haotian Mai, Fuming Fang, Huy H. Nguyen, Junichi Yamagishi, and Isao Echizen. 2019.  Generating sentiment-preserving fake online reviews using neural language models and their human- and machine-based detection.
  * Blitzstein and Hwang (2015) Joseph K. Blitzstein and Jessica Hwang. 2015.  _Introduction to Probability_.  CRC Press.
  * Chakraborty et al. (2023) Souradip Chakraborty, Amrit Singh Bedi, Sicheng Zhu, Bang An, Dinesh Manocha, and Furong Huang. 2023.  On the possibilities of ai-generated text detection.
  * Clark et al. (2021) Elizabeth Clark, Tal August, Sofia Serrano, Nikita Haduong, Suchin Gururangan, and Noah A. Smith. 2021.  All that’s ’human’ is not gold: Evaluating human evaluation of generated text.
  * Crothers et al. (2022) Evan Crothers, Nathalie Japkowicz, and Herna Viktor. 2022.  Machine generated text: A comprehensive survey of threat models and detection methods.
  * Dehouche (2021) Nassim Dehouche. 2021.  Plagiarism in the age of massive generative pre-trained transformers (gpt-3): “the best time to act was yesterday. the next best time is now.”.  _Ethics in Science and Environmental Politics_ , 21.
  * Dey et al. (2023) Nolan Dey, Daria Soboleva, Faisal Al-Khateeb, Bowen Yang, Ribhu Pathria, Hemant Khachane, Shaheer Muhammad, Zhiming, Chen, Robert Myers, Jacob Robert Steeves, Natalia Vassilieva, Marvin Tom, and Joel Hestness. 2023.  Btlm-3b-8k: 7b parameter performance in a 3b parameter model.
  * Fagni et al. (2021) Tiziano Fagni, Fabrizio Falchi, Margherita Gambini, Antonio Martella, and Maurizio Tesconi. 2021.  TweepFake: About detecting deepfake tweets.  _PLOS ONE_ , 16(5):e0251415.
  * Fan et al. (2018) Angela Fan, Mike Lewis, and Yann Dauphin. 2018.  Hierarchical neural story generation.
  * Gao et al. (2022) Tianyu Gao, Xingcheng Yao, and Danqi Chen. 2022.  Simcse: Simple contrastive learning of sentence embeddings.
  * Gehrmann et al. (2019) Sebastian Gehrmann, Hendrik Strobelt, and Alexander M. Rush. 2019.  GLTR: statistical detection and visualization of generated text.  _CoRR_ , abs/1906.04043.
  * Guo et al. (2023) Biyang Guo, Xin Zhang, Ziyuan Wang, Minqi Jiang, Jinran Nie, Yuxuan Ding, Jianwei Yue, and Yupeng Wu. 2023.  How close is chatgpt to human experts? comparison corpus, evaluation, and detection.
  * Hoffmann et al. (2022) Jordan Hoffmann, Sebastian Borgeaud, Arthur Mensch, Elena Buchatskaya, Trevor Cai, Eliza Rutherford, Diego de Las Casas, Lisa Anne Hendricks, Johannes Welbl, Aidan Clark, Tom Hennigan, Eric Noland, Katie Millican, George van den Driessche, Bogdan Damoc, Aurelia Guy, Simon Osindero, Karen Simonyan, Erich Elsen, Jack W. Rae, Oriol Vinyals, and Laurent Sifre. 2022.  Training compute-optimal large language models.
  * Kirchenbauer et al. (2023a) John Kirchenbauer, Jonas Geiping, Yuxin Wen, Jonathan Katz, Ian Miers, and Tom Goldstein. 2023a.  A watermark for large language models.
  * Kirchenbauer et al. (2023b) John Kirchenbauer, Jonas Geiping, Yuxin Wen, Manli Shu, Khalid Saifullah, Kezhi Kong, Kasun Fernando, Aniruddha Saha, Micah Goldblum, and Tom Goldstein. 2023b.  On the reliability of watermarks for large language models.
  * Krishna et al. (2023) Kalpesh Krishna, Yixiao Song, Marzena Karpinska, John Wieting, and Mohit Iyyer. 2023\.  Paraphrasing evades detectors of ai-generated text, but retrieval is an effective defense.
  * Li et al. (2023) Xiang Lisa Li, Ari Holtzman, Daniel Fried, Percy Liang, Jason Eisner, Tatsunori Hashimoto, Luke Zettlemoyer, and Mike Lewis. 2023.  Contrastive decoding: Open-ended text generation as optimization.
  * Liang et al. (2023) Weixin Liang, Mert Yuksekgonul, Yining Mao, Eric Wu, and James Zou. 2023.  Gpt detectors are biased against non-native english writers.
  * Lin et al. (2022) Stephanie Lin, Jacob Hilton, and Owain Evans. 2022.  TruthfulQA: Measuring how models mimic human falsehoods.  In _Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 3214–3252, Dublin, Ireland. Association for Computational Linguistics.
  * Merity et al. (2016) Stephen Merity, Caiming Xiong, James Bradbury, and Richard Socher. 2016.  Pointer sentinel mixture models.
  * Mitchell et al. (2023) Eric Mitchell, Yoonho Lee, Alexander Khazatsky, Christopher D. Manning, and Chelsea Finn. 2023.  Detectgpt: Zero-shot machine-generated text detection using probability curvature.
  * Pan et al. (2023) Yikang Pan, Liangming Pan, Wenhu Chen, Preslav Nakov, Min-Yen Kan, and William Yang Wang. 2023.  On the risk of misinformation pollution with large language models.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Köpf, Edward Yang, Zach DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  Pytorch: An imperative style, high-performance deep learning library.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.
  * Sadasivan et al. (2023) Vinu Sankar Sadasivan, Aounon Kumar, Sriram Balasubramanian, Wenxiao Wang, and Soheil Feizi. 2023.  Can ai-generated text be reliably detected?
  * Sanh et al. (2020) Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. 2020.  Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter.
  * Solaiman et al. (2019) Irene Solaiman, Miles Brundage, Jack Clark, Amanda Askell, Ariel Herbert-Voss, Jeff Wu, Alec Radford, Gretchen Krueger, Jong Wook Kim, Sarah Kreps, Miles McCain, Alex Newhouse, Jason Blazakis, Kris McGuffie, and Jasmine Wang. 2019.  Release strategies and the social impacts of language models.
  * Touvron et al. (2023) Hugo Touvron, Louis Martin, Kevin Stone, Peter Albert, Amjad Almahairi, Yasmine Babaei, Nikolay Bashlykov, Soumya Batra, Prajjwal Bhargava, Shruti Bhosale, Dan Bikel, Lukas Blecher, Cristian Canton Ferrer, Moya Chen, Guillem Cucurull, David Esiobu, Jude Fernandes, Jeremy Fu, Wenyin Fu, Brian Fuller, Cynthia Gao, Vedanuj Goswami, Naman Goyal, Anthony Hartshorn, Saghar Hosseini, Rui Hou, Hakan Inan, Marcin Kardas, Viktor Kerkez, Madian Khabsa, Isabel Kloumann, Artem Korenev, Punit Singh Koura, Marie-Anne Lachaux, Thibaut Lavril, Jenya Lee, Diana Liskovich, Yinghai Lu, Yuning Mao, Xavier Martinet, Todor Mihaylov, Pushkar Mishra, Igor Molybog, Yixin Nie, Andrew Poulton, Jeremy Reizenstein, Rashi Rungta, Kalyan Saladi, Alan Schelten, Ruan Silva, Eric Michael Smith, Ranjan Subramanian, Xiaoqing Ellen Tan, Binh Tang, Ross Taylor, Adina Williams, Jian Xiang Kuan, Puxin Xu, Zheng Yan, Iliyan Zarov, Yuchen Zhang, Angela Fan, Melanie Kambadur, Sharan Narang, Aurelien Rodriguez, Robert Stojnic, Sergey Edunov, and Thomas Scialom. 2023.  Llama 2: Open foundation and fine-tuned chat models.
  * Ueoka et al. (2021) Honai Ueoka, Yugo Murawaki, and Sadao Kurohashi. 2021.  Frustratingly easy edit-based linguistic steganography with a masked language model.
  * Vaswani et al. (2023) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Lukasz Kaiser, and Illia Polosukhin. 2023.  Attention is all you need.
  * Weidinger et al. (2021) Laura Weidinger, John Mellor, Maribeth Rauh, Conor Griffin, Jonathan Uesato, Po-Sen Huang, Myra Cheng, Mia Glaese, Borja Balle, Atoosa Kasirzadeh, Zac Kenton, Sasha Brown, Will Hawkins, Tom Stepleton, Courtney Biles, Abeba Birhane, Julia Haas, Laura Rimell, Lisa Anne Hendricks, William S. Isaac, Sean Legassick, Geoffrey Irving, and Iason Gabriel. 2021.  Ethical and social risks of harm from language models.  _CoRR_ , abs/2112.04359.
  * Wieting et al. (2023) John Wieting, Kevin Gimpel, Graham Neubig, and Taylor Berg-Kirkpatrick. 2023.  Paraphrastic representations at scale.
  * Wolf et al. (2019) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Rémi Louf, Morgan Funtowicz, et al. 2019.  Huggingface’s transformers: State-of-the-art natural language processing.  _arXiv preprint arXiv:1910.03771_.
  * Zellers et al. (2019) Rowan Zellers, Ari Holtzman, Hannah Rashkin, Yonatan Bisk, Ali Farhadi, Franziska Roesner, and Yejin Choi. 2019.  Defending against neural fake news.  _CoRR_ , abs/1905.12616.
  * Zhang et al. (2022) Susan Zhang, Stephen Roller, Naman Goyal, Mikel Artetxe, Moya Chen, Shuohui Chen, Christopher Dewan, Mona Diab, Xian Li, Xi Victoria Lin, Todor Mihaylov, Myle Ott, Sam Shleifer, Kurt Shuster, Daniel Simig, Punit Singh Koura, Anjali Sridhar, Tianlu Wang, and Luke Zettlemoyer. 2022.  Opt: Open pre-trained transformer language models.
  * Zhao et al. (2023) Xuandong Zhao, Yu-Xiang Wang, and Lei Li. 2023.  Protecting language generation models via invisible watermarking.  _arXiv preprint arXiv:2302.03162_.

##  Appendix A Sample Generated Texts

prompt |  real completion |  no watermark |  Maryland watermarked |  Sampling Watermarked(WR) |  Sampling Watermarked(WOR)
---|---|---|---|---|---
…Deputy Short radioed for back-up, drew his firearm and commanded Zubrod to step back from the victim. Zubrod initially complied by stepping back and dropping the hammer. Deputy Short holstered his gun and took out his |  TASER. Zubrod responded by saying something about finding a gun and began to reach down under the bed. He came up empty-handed, grabbed a nearby pair of scissors and immediately stabbed Schukei in the neck while she was lying on the floor.\n Deputy Short fired his TASER but it was not effective in incapacitating Zubrod[…continues] |  handcuffs. As he turned to go back to the house, Zubrod came at him and punched him in the face.\n Deputy Short, who was not wearing a bulletproof vest, fired his firearm, striking Zubrod in the chest and killing him[…continues] |  handcuffs. \n When Deputy Short was within reach, Zubrod suddenly swung the hammer and hit him on the side of the head, causing severe injury to the deputy. Deputy Short was hospitalized for five days, underwent a number of surgeries, including a craniotomy, and suffered extensive brain trauma.[…continues] |  taser and pointed it at Zubrod. Deputy Short did not know that the hammer had been de-cocked at that point. Deputy Short later claimed that the hammer was de-cocked. Deputy Short claimed that his firearm discharged once in the bedroom area, once in the bathroom area[…continues] |  patrol binder, direct traffic from a motorized ATV with no lights on or turn-on indicator. Deputy Layne Stuckey responds to assist deputize the situation and back Deputee’Short off from continuing the investigation. Deputee’ Stubke asks where I’m from, I respond the “Same City”. […continues]
…They didn’t say all wildlife management is based on proven science, not guesswork and theories by people who oppose hunting. Thirty-three states have bear hunting, and Missouri will probably have bear hunting when the bear population reaches 500.\n New Jersey bear population is estimated at about 2, |  400, probably more than that from the cubs born last January. The most densely state–ours–and the second smallest in size.\n And the Show-Me State with 68,700 square miles worries about bears when the population reaches 500! New Jersey has 8,722 square miles with too many bears.\n New Jersey hunters never demanded bear hunting.[…continues] |  000, and the bear hunt is set for Oct. 12-14.\n The antihunters don’t want the hunt, even though the bear population is growing, and they want to put the state’s bear population on a diet.\n The state has about 2,000 acres of land set aside for the bear population, and the antihunters want to make it larger.[…continues] |  000\. \n This year’s bear hunting will be a lottery.\n Antihunters said it’s cruel to kill bears.\n So it is cruel to kill animals that can’t kill us, like mice, deer, geese, turkeys, etc. \n Antihunters said it’s cruel to kill animals that can’t kill us, like mice, deer, geese, turkeys, etc. \n Bears kill people.[…continues] |  500-3,000 bears.\n In Missouri, 11% of the bear population was taken last year.\n And there have been no cases this fall that I know of of people getting mauaued by bears.\n In their rush for an antihunting victory, they are hurting the bears.[…continues] |  512.\n Think that is too low a number for this size area for beer. A biased count was done by the DFWA (Don Foschi Alpha Male Animals), whose members like Fisk (Fox-Skate or Fox-Guess). Guesses based on a low population. And what do they propose, an airplane flyoever to count.[…continues]
…It passed 68-30, with a group of 27 Democrats and one independent, mostly from states where gun rights are widely supported, joining all 40 Republicans in voting for the measure.\n Under the Wicker amendment, |  Amtrak would lose the funds earmarked for it in the must-pass spending bill if it did not comply with the new regulations. In a statement released after the amendment passed, Mr. Wicker stressed that the guidelines laid out in the provision[…continues] |  the federal government would be prohibited from withholding federal grants from states that do not allow people with concealed-carry permits to carry their guns across state lines.\n The amendment, which was attached to a $494 billion spending bill for […continues] |  the National Park Service, the Fish and Wildlife Service and the Bureau of Land Management would be banned from enforcing any federal regulation “relating to the regulation of the use, possession, transfer, receipt, shipment or surplus of any firearm, ammunition[…continues] |  …18 to 20 gun shows held year-round by fairgrounds or nonprofit shooting clubs in major cities would no longer have to require a special license.\n Those currently required are in places like Staten Island, N.Y.; Portland, Ore.; and Birmingham, Ala. And they are often required for only certain days of events.[…continues] |  guns could also not law enforcers to keep guns with gun-restriction laws and rules impende.\n A version of this article first published September27. Read it on npr org. Photo Credite. Nicolette Barr. Nicolette spent a week on safat.[…continues]
…Saturday’s 2-2 draw with Oxford United saw Barton’s 10th-placed team switch from 4-4-2 to 4-3-3, the formation he says he wants to play from now on as Town prepare to travel |  to Rochdale this weekend.\n The head coach said: “The key for me is how do we progress and move forward. We changed system on Saturday.\n “This is how we will be playing going forward. I think it is the correct way to play.[…continues] |  to Portsmouth on Saturday.\n Barton said: “It’s here to stay. We’ll look to play that formation for the rest of the season.\n “I think we’ve got the players to play that way.\n “We’ve got a good squad. We’ve got a good mix of youth and experience.[…continues] |  to Oldham Athletic on Tuesday.\n Town were two goals up in the first half before dropping off and allowing Oxford to fight back and, when the visitors went in 2-2, it felt like it might have been a bad idea.\n Barton was disappointed that Town[…continues] |  to Plymouth Argyle in League One on Saturday.\n Former central midfield partners Ephron Mason-Randolph and Lewis Evans were shifted to wide roles on the right hand-side and former captain Jimmy-Bell found his place, too, as he[…continues] |  across to Blackwell Mewse on January.\n And in the first instance Town lined-up in five changes as Tom Lawrence was brought in and Ben Lindsard restored and both Craig Mackrell - after ankledown injury[…continues]
Table 1A: Some generation samples for watermarks. Model:Llama-2-7B,
Dataset:c4, $y$:5, $k$:1, Temperature:1.0


# Linking Langevin equation to scaling properties of space plasma turbulence
at sub-ion scales

Simone Benella1, Mirko Stumpo1, Tommaso Alberti2, Oreste Pezzi3,1, Emanuele
Papini1, Emiliya Yordanova4, Francesco Valentini5 and Giuseppe Consolini1
1Istituto di Astrofisica e Planetologia Spaziali, Istituto Nazionale di
Astrofisica, via del Fosso del Cavaliere 100, I-00133, Rome, Italy
2Istituto Nazionale di Geofisica e Vulcanologia, Via di Vigna Murata, 605,
I-00143, Rome, Italy
3Istituto per la Scienza e Tecnologia dei Plasmi, Consiglio Nazionale delle
Ricerche, Via Amendola 122/D, I-70126 Bari, Italy
4Swedish Institute for Space Physics, Uppsala, Sweden
5Dipartimento di Fisica, Università della Calabria, Ponte P. Bucci, Cubo 31C,
I-87036 Arcavacata di Rende (CS), Italy

###### Abstract

Current understanding of the kinetic-scale turbulence in weakly-collisional
plasmas still remains elusive. We employ a general framework in which the
turbulent energy transfer is envisioned as a scale-to-scale Langevin process.
Fluctuations in the sub-ion range show a global scale invariance, thus
suggesting a homogeneous energy repartition. In this Letter, we interpret such
a feature by linking the drift term of the Langevin equation to scaling
properties of fluctuations. Theoretical expectations are verified on solar
wind observations and numerical simulations thus giving relevance to the
proposed framework for understanding kinetic-scale turbulence in space
plasmas.

Space and astrophysical plasmas, such as the solar and stellar winds, the
planetary magnetospheres, and the interstellar medium, often exhibit a
strongly turbulent dynamics. The energy of fluctuations cross-scale transfers
over a vast range of spatial and temporal scales and diverse structures,
including current sheets, magnetic islands, and vortices, emerge. Since about
sixty years, a large fleet of spacecraft has been providing in-situ
measurements of the solar wind plasma and the interplanetary magnetic field
Bruno and Carbone (2016); Chasapis  _et al._ (2017). These observations have
allowed us to investigate in great detail turbulence in space plasmas, whose
study is of pivotal importance also for clarifying the dynamics of far
astrophysical objects not accessible by in-situ spacecraft.

Owing to their weak collisionality, understanding the properties of turbulence
at scales similar or smaller than the inertial ion scales, henceforth referred
as sub-ion scales, is essential for clarifying the energy transfer and
dissipation mechanisms in space and astrophysical plasmas. Indeed, at sub-ion
scales, the solar wind dynamics is a complex puzzle composed of several
intertwined plasma processes. Such processes generally manifest in the entire
phase space, since inter-particle collisions are infrequent and plasmas can be
far from the thermal equilibrium (Cassak  _et al._ , 2023).

Historically, the investigation of plasma dynamics at sub-ion scales has aimed
at identifying the plasma processes successfully explaining in-situ
observations. These include kinetic-scale fluctuations (e.g., kinetic-Alfvén
and whistler waves) Alexandrova  _et al._ (2008); Sahraoui  _et al._ (2009);
Chen  _et al._ (2010, 2013), microinstabilities Matteini  _et al._ (2012),
magnetic reconnection Zenitani  _et al._ (2011); Wan  _et al._ (2015), energy
transfer highlighted by pressure-strain interactions Bandyopadhyay  _et al._
(2020), Landau damping Chen  _et al._ (2019) etc., (see Verscharen  _et al._ ,
2019, for a review). These mechanisms often depend on different plasma
parameters (e.g., the plasma $\beta$) and, hence, their applicability is
specialized to specific plasma conditions. Moreover, some of these mechanisms,
such as kinetic-scale fluctuations, microinstabilities, and Landau damping,
are often introduced within a linear or quasi-linear framework, thus possibly
making their application less suitable in cases of strong turbulence, as
observed in the solar wind Matthaeus  _et al._ (2014).

A complementary and additional perspective proposes to interpret space plasma
turbulence through a small set of macroscopic variables, whose fluctuations
are not negligible but rather control the system dynamics Sekimoto (2010).
Such a definition sets the ground for a general and interdisciplinary
framework whose underlying idea is that an observed coarse-grained dynamics
can be modeled as a stochastic processes. A stunning example of this approach
is represented by hydrodynamic turbulence, where longitudinal velocity
fluctuations at different scales can be described by means of a stochastic
approach (e.g., Faranda  _et al._ , 2017; Dubrulle, 2019; Alberti  _et al._ ,
2023), mostly based on the generalized Langevin equation Friedrich and Peinke
(1997); Davoudi and Rahimi Tabar (1999); Renner  _et al._ (2001a); Milan  _et
al._ (2013); Nickelsen and Engel (2013); Fuchs  _et al._ (2022). From this
point of view, the scale-by-scale evolution of the probability distribution
functions (PDFs) of the longitudinal increments is governed by the Fokker-
Planck equation (FPE). A similar description has been also applied to
turbulence in space plasmas. An earlier work by Strumik and Macek (2008)
showed that the inertial-range cascade of the solar wind can be properly
framed in this scenario. A step forward has been made by proving that this
formalism applies to sub-ion scales, thanks to unprecedented high-resolution
magnetic field observations Benella  _et al._ (2022a, b); Macek  _et al._
(2023). These studies confirm that the FPE is accurate in reproducing the
observed PDFs of magnetic field fluctuations, and show that PDFs outline a
globally self-similar scenario Kiyani  _et al._ (2009); Osman  _et al._
(2015); Benella  _et al._ (2022a, b); Macek  _et al._ (2023).

A common way to investigate solar wind turbulence is to analyze high-order
statistics of magnetic field fluctuations, viz., the structure functions
$S^{(q)}_{r}=\langle b_{r}^{q}\rangle$, where

$b_{r}\doteq B(x+r)-B(x)$ (1)

are the increments of the magnetic field $B(x)$ at the spatial scale $r$.
Fluctuations $b_{r}$ are globally scale invariant if

$r\to\lambda r\Rightarrow b_{\lambda r}=\lambda^{h}b_{r},$ (2)

where $h$ is a scaling exponent and $\lambda\in\mathbb{R}^{+}$ is a
dilatation/contraction parameter representing a zoom-in/out procedure through
magnetic field structures. The basic idea is that each structure at a scale
decays into smaller-scale structures in a scale-invariant way. Indeed, Eq. (2)
implies that structure functions exhibit a power-law trend as a function of
$r$, i.e., $S^{(q)}_{r}\sim r^{\zeta_{q}}$, with the scaling exponents
$\zeta_{q}$. The space-time distribution of turbulent structures reflects the
properties of energy dissipation rate and related underlying processes Frisch
(1995). However, there are no fully deductive theories that are able to derive
the scaling laws, viz., the behavior of the scaling exponents, starting from
dynamic equations. For this reason, phenomenological theories, which introduce
additional hypothesis based on experimental evidences, are widely used. In
this context, the method described in this Letter ranks as a novel
phenomenological perspective of sub-ion scale plasma turbulence.

The transformation introduced in Eq. (2) is satisfied by $b_{r}\sim r^{h}$
Frisch (1995). In the multifractal theory of magnetohydrodynamic (MHD)
turbulence, $h$ is the generalized Hölder exponent $h(x,r)$ Hardy (1916) and
thus magnetic field fluctuations are locally scale-invariant, $b_{r}\sim
r^{h(x,r)}$. Here, $h(x,r)$ represents a fluctuating quantity Landau and
Lifshitz (2013), hence the global scale-invariance holds only on average
Frisch (1995) and can be expressed in terms of the most probable Hölder
exponent $h_{0}$. For a globally scale-invariant dynamics, a relation between
$h_{0}$ and the spectral slope $\beta$ of the Fourier power spectral density
can be derived

$\beta=2h_{0}+1,$ (3)

being $\beta\in(1,3)$ for the non-stationary process $B(x)$ with stationary
increments $b_{r}$ Flandrin (1989).

In terms of FPE, the scale-evolution equation of the $q$-th order structure
function can be written as Benzi  _et al._ (1993); Renner  _et al._ (2001b);
Arnèodo  _et al._ (2008)

$-\frac{\partial}{\partial r}S^{(q)}_{r}=q\langle
b_{r}^{q-1}D^{(1)}(b_{r})\rangle+q(q-1)\langle
b_{r}^{q-2}D^{(2)}(b_{r})\rangle.$ (4)

This scale-evolution equation depends on the two Kramers-Moyal (KM)
coefficients which define the stochastic process. For the solar wind, the
following parameterization of the KM coefficients correctly reproduce the
fluctuation statistics at sub-ion scales Benella  _et al._ (2022a, b); Macek
_et al._ (2023):

$\displaystyle D^{(1)}(b_{r})=-\gamma(r)\,b_{r}$ (5) $\displaystyle
D^{(2)}(b_{r})=\alpha(r)+\delta(r)\,b_{r}^{2}.$

By inserting Eqs. (5) in Eq. (4), a direct relation between KM coefficient
parameters and the local scaling exponent $\xi_{q}(r)$ can be drawn Renner
_et al._ (2001b); Reinke  _et al._ (2018):

$\xi_{q}(r)=\frac{r}{S^{(q)}_{r}}\frac{\partial S^{(q)}_{r}}{\partial
r}=rq\biggl{(}\gamma(r)+(1-q)\biggl{(}\delta(r)+\frac{S^{(q-2)}_{r}}{S^{(q)}_{r}}\alpha(r)\biggr{)}\biggr{)}.$
(6)

In the framework of our model, it means that the evolution in scale of the
fluctuating trajectories $b_{r}$ can be described according with the Langevin
equation Risken (1996):

$-\frac{\partial b_{r}}{\partial
r}=D^{(1)}(b_{r},r)+\sqrt{2D^{(2)}(b_{r},r)}\eta(r),$ (7)

where $\eta(r)$ is a unit-variance Gaussian noise term such that
$\langle\eta(r)\eta(r^{\prime})\rangle=\delta(r-r^{\prime})$. Eq. (7)
indicates that the dynamics of magnetic field increments as a function of the
scale $r$ is the result of the superposition of two mechanisms: the drift
which models the self-similar decay of magnetic field structures from larger
to smaller ones and the diffusion that accounts for the stochastic repartition
of energy during the structure breakdown process.

This scenario applies in the general framework of turbulence. In the following
we specialize our discussion to the sub-ion scale dynamics by considering
three different data samples: super-Alfvénic solar wind observations gathered
by PSP during its first perihelion Chhiber  _et al._ (2021), a high-speed
stream observed by the ESA/Cluster mission in the near-Earth solar wind
Alberti  _et al._ (2019), and a data sample obtained from Eulerian Hybrid
Vlasov-Maxwell (HVM) simulations of decaying plasma turbulence Pezzi  _et al._
(2019, 2021). The HVM numerical framework has been widely adopted to describe
significant features of turbulence in space plasmas (Servidio  _et al._ ,
2015; Cerri  _et al._ , 2017; Franci  _et al._ , 2018), including the
transition to a monofractal scaling when approaching sub-ion scales (Leonardis
_et al._ , 2016).

Figure 1: Fluctuating magnetic field trajectories normalized to the standard
deviation scale-by-scale from the PSP data sample (color lines). The inset
shows the averaged Hölder exponent $\overline{h}(r)$ as a function of $r$
(solid line) along with 95% confidence bounds (dashed lines) for the PSP data
sample. Vertical lines indicate the ion inertial length.

The data selected for the analysis show a similar level of fluctuation
$b_{\text{rms}}/B_{0}\sim 0.5$, being $B_{0}$ the mean magnetic field, and
different values of the proton plasma beta (0.4 for PSP, 1.5 for Cluster and 2
for HVM). The analysis of spacecraft measurements is carried out by assuming
the Taylor’s hypothesis: $r=\langle V\rangle\tau$, where $\tau$ indicates the
time scale and $\langle V\rangle$ the mean solar wind speed Narita  _et al._
(2013); Chhiber  _et al._ (2019). Plasma turbulence in the solar wind is known
to be strongly anisotropic and the energy cascade develops mostly in the
transverse directions with respect to the mean local magnetic field Shebalin
_et al._ (1983); Matthaeus  _et al._ (1996); Verma (2004); Matthaeus  _et al._
(2012). Hence, we analyze fluctuations along one of the transverse magnetic
field directions for each data sample considered here. For in-situ
observations we compute increments along the maximum variance component
$b_{r}=b_{r,\max}$, while in the case of HVM simulations we consider
increments along the $x$-component $b_{r}=b_{r,x}$ since the mean field is
directed along the $z$-direction Pezzi  _et al._ (2019).

The first step in our reasoning consists in assessing the different role
played by the deterministic and the stochastic terms in Eq. (7) at different
scales. To this purpose we define a set of trajectories $b_{r}$, see Eq. (1),
for different starting points $x$. For a nearly constant value of the Hölder
exponent, say $h_{0}$, the amplitude of fluctuations is expected to decrease
as a power law in $r$. If we normalize each trajectory by the standard
deviation $\sigma_{b_{r}}$, which then scales as $r^{h_{0}}$, we expect to
observe constant and scale-independent values of the normalized fluctuations.
In this view, Figure 1 reports a few normalized trajectories which allow us to
identify two different dynamical regimes: i) the sub-ion range where
normalized fluctuations show a nearly constant value; ii) the lower end of the
inertial range, e.g., at scales $\gtrsim 10d_{i}$, where trajectories exhibit
stochastic fluctuations. The different level of stochasticity of the magnetic
field fluctuations at different scales can be quantified by estimating the
local Hölder exponent. The inset of Figure 1 reports the scale-by-scale
averaged Hölder exponent $\overline{h}(r)$ (solid line) along with the 95%
confidence interval bounds (dashed lines). The stochastic character of
turbulent fluctuations is related to the spreading of the local Hölder
exponent (i.e., spreading of the confidence bounds) when approaching the
inertial range ($r>10d_{i}$). In the context of Langevin equation, this
scenario suggests that the diffusion term does not significantly contribute to
the dynamics at sub-ion scales, whereas it plays an important role in the
inertial range. Results of Figure 1 are derived from PSP data but, analogous
considerations can be also extended to the other data samples (not shown).
Based on this observation, we introduce the non-diffusive limit of the
Langevin equation in order to model sub-ion scale fluctuations.

Hence, we assume that the small-scale dynamics is governed only by the first-
order KM coefficient, viz., $D^{(1)}\gg D^{(2)}$, which in terms of the
parametrization introduced in Eqs. (5) corresponds to neglecting the
parameters $\alpha(r)$ and $\delta(r)$. Furthermore, $\gamma(r)$ follows a
power-law behavior at sub-ion scales Benella  _et al._ (2022a, b); Macek  _et
al._ (2023), i.e.,

$\gamma(r)=h_{0}r^{\mu},$ (8)

allowing us to derive the coefficients $h_{0}$ and the exponents $\mu$
directly from data. Their values are listed in Table 1. In the non-diffusive
limit, Eq. (6) becomes

$\xi_{q}(r)\sim rq\,\gamma(r)=qh_{0}r^{1+\mu}.$ (9)

For both solar wind data and numerical simulations $\mu\sim-1$ (see Table 1),
then Eq. (9) reads

$\zeta_{q}\equiv\xi_{q}(r)=qh_{0},$ (10)

where $h_{0}$, in this fashion, corresponds to the most probable Hölder
exponent. This means that the condition $D^{(1)}\gg D^{(2)}$ implies a global
scale-invariance. This represents the central result of this work since it
establishes a direct link between the drift term at sub-ion scales and the
most probable Hölder exponent of the multifractal theory. Such a connection
will be now verified by exploiting both in-situ observations and numerical
simulations in a two-fold sense: on a statistical level, e.g., through
structure functions, and on an individual level, e.g., by considering single
fluctuating trajectories.

Scaling exponents can be derived from solar wind observations through the
structure function analysis within the sub-ion range.The scaling exponents
obtained from the three data samples are here compared with those obtained
through Eq. (10), Figure 2. An excellent agreement between observations and
model predictions is found, strongly supporting the established link between
the first-order KM coefficient and the Hölder exponent $h_{0}$. Concerning the
spectral analysis, we can use Eq. (3) to estimate the spectral slope
corresponding to the observed most probable Hölder exponent $h_{0}$ from the
three data samples. By using the values estimated from the drift coefficient,
reported in Table 1, we obtain $\beta\sim[2.7,2.8]$, which is in agreement
with typical findings in the sub-ion range Sahraoui  _et al._ (2013); Carbone
_et al._ (2022).

| PSP | Cluster | HVM
---|---|---|---
$h_{0}$ | $0.87\pm 0.06$ | $0.87\pm 0.03$ | $0.88\pm 0.52$
$\mu$ | $-1.02\pm 0.01$ | $-1.04\pm 0.01$ | $-1.04\pm 0.22$
Table 1: Fitted $h_{0}$ and $\mu$ parameters appearing in Eq. (8). Errors
represent fit 95% confidence bounds. Figure 2: Scaling exponents $\zeta_{q}$
as a function of the order $q$ of the structure functions (markers). Straight
lines represent the estimation provided by Eq. (10) by using the values of
Table 1. Figure 3: Comparison between three trajectories selected from
different data samples (left panel) and the corresponding non-diffusive
trajectories $b_{r}^{*}$ (black dotted lines). Root mean square error of the
fluctuations $b_{r}$ with respect to $b_{r}^{*}$ as a function of the scale
(right panel). The red dotted line indicates a reference power-law trend. The
scales are reported in units of $d_{i}$ and the magnetic field is normalized
by the mean field value $B_{0}$ for clarity of presentation.

The linear trend of the scaling exponents $\zeta_{q}$ as a function of the
order of the structure function $q$ observed from data suggests that magnetic
field fluctuations should exhibit a weak stochastic behavior at sub-ion
scales. In fact, the non-diffusive limit of Eq. (7) corresponds to a damping
equation with a scale dependent drift term

$\frac{\partial b_{r}^{*}}{\partial
r}=\gamma(r)b_{r}^{*}=\frac{h_{0}b_{r}^{*}}{r},$ (11)

whose solution is:

$b_{r}^{*}=b_{0}\biggl{(}\frac{r}{r_{0}}\biggr{)}^{h_{0}},$ (12)

where $b_{r}^{*}$ indicates the deterministic solution and $b_{0}$ is the
initial condition, i.e., a value of the fluctuation at the scale $r_{0}$. The
left panel of Figure 3 shows a comparison of the power-law damping predicted
by Eq. (12) with a set of fluctuating trajectories from three different data
samples. This figure points out that the non-diffusive approximation is quite
satisfactory for $r\ll d_{i}$, whereas the diffusion starts to significantly
affect the dynamics of the magnetic field fluctuations at $r\gtrsim d_{i}$.
The root mean square error between the whole set of measured trajectories and
the corresponding model predictions is reported in Figure 3 (right panel).
This figure shows that errors decrease more rapidly when approaching sub-ion
scales for all three data samples.

The proposed approach allows us to unveil connections between the global
statistical features of the plasma dynamics and the peculiar characteristics
of each realization of the cascading process. For instance, the Hölder
exponent $h_{0}$ quantifies the regularity (i.e., the roughness) of the
topology of the fluctuations associated with a dynamical process Nguyen  _et
al._ (2020): irregular bursts are characterized by $h_{0}\to 0$, whereas a
field of increasingly regular fluctuations is associated with larger $h_{0}$
and leans towards an homogeneous fractal structure when $h_{0}\to 1$. By
looking at Figure 3 we can observe that the deterministic-like damping shown
by empirical fluctuating trajectories is characterized by a weakly fluctuating
trend that corresponds to a well-defined sign of the fluctuation amplitude
among different scales. The high values obtained for $h_{0}\sim[0.87,0.88]$
represent a natural consequence of this intuitive framework.

In the case of the Langevin equation, the non-diffusive limit is achieved by
simply neglecting the noise term and then reducing the stochastic equation to
a deterministic one. The same procedure cannot be applied when dealing with
the FPE, for which the non-diffusive limit is singular Gardiner  _et al._
(1985); van Kampen (1992). In this case the correct way to perform this limit
is to introduce a new variable $y$ such that $b_{r}=b^{*}_{r}+\epsilon y$,
where $\epsilon$ is an arbitrarily small parameter. Hence, the non-diffusive
limit corresponds to $\epsilon\to 0$ in which case the stochastic fluctuation
vanishes and $b_{r}$ approaches the deterministic solution $b_{r}^{*}$. This
approximation represents an ideal limit in which the FPE predicts weak
fluctuations around the deterministic damping van Kampen (1992). Such scenario
seems to be fairly accurate by inspecting Figure 3, where the trajectories
exhibit weak fluctuations around the asymptotic solution at small scales.

All the evidences reported in this work are consistent with a homogeneous and
globally self-similar repartition of energy through the scales in the sub-ion
range. As a consequence, the typical multifractal character of the non-
homogeneuous energy repartition observed in the inertial range, here enclosed
in the diffusion term of the Langevin equation, does not represent a major
contribution at these scales. The result is an unstructured fluctuation field
suggesting that typical space-time inhomogeneities associated with turbulent
structures have sizes larger than $d_{i}$. This is supported by the linear
scaling law of $\zeta_{q}$, here derived in the non-diffusive limit, and by
the fact that trajectories at sub-ion scales are captured with a single
exponent $h_{0}$. In this context, our newly introduced framework constitutes
a complementary view with respect to the existing phenomenological models,
allowing us to recover important results of solar wind turbulence at sub-ion
scales and to provide new constraints for future model developments.

###### Acknowledgements.

This work is supported by the mini-grant “The solar wind: a paradigm for
complex system dynamics” financed by the National Institute for Astrophysics
under the call “Fundamental Research 2022”. E.Y. work was funded by the
Swedish Space Agency, grant 192/20. PSP data used in this study are available
at the NASA Space Physics Data Facility (SPDF),
https://spdf.gsfc.nasa.gov/index.html. The authors acknowledge the
contributions of the FIELDS team to the Parker Solar Probe mission. The
authors acknowledge the Cluster FGM and STAFF P.I.s and teams and the ESA-
Cluster Science Archive for making available the data used in this work. The
FGM and STAFF data were combined using the IRFU-Matlab analysis package
available at https://github.com/irfu/irfu-matlab.

## References

  * Bruno and Carbone (2016) R. Bruno and V. Carbone,  _Turbulence in the Solar Wind_, Vol. 928 (2016).
  * Chasapis  _et al._ (2017) A. Chasapis, W. H. Matthaeus, T. N. Parashar, S. A. Fuselier, B. A. Maruca, T. D. Phan, J. L. Burch, T. E. Moore, C. J. Pollock, D. J. Gershman, R. B. Torbert, C. T. Russell, and R. J. Strangeway, High-resolution Statistics of Solar Wind Turbulence at Kinetic Scales Using the Magnetospheric Multiscale Mission, The Astrophysical Journal Letters 844, L9 (2017).
  * Cassak  _et al._ (2023) P. A. Cassak, M. H. Barbhuiya, H. Liang, and M. R. Argall, Quantifying Energy Conversion in Higher-Order Phase Space Density Moments in Plasmas, Physical Review Letters 130, 085201 (2023).
  * Alexandrova  _et al._ (2008) O. Alexandrova, V. Carbone, P. Veltri, and L. Sorriso-Valvo, Small-Scale Energy Cascade of the Solar Wind Turbulence, Astrophysical Journal 674, 1153 (2008), arXiv:0710.0763 [astro-ph] .
  * Sahraoui  _et al._ (2009) F. Sahraoui, M. L. Goldstein, P. Robert, and Y. V. Khotyaintsev, Evidence of a Cascade and Dissipation of Solar-Wind Turbulence at the Electron Gyroscale, Physical Review Letters 102, 231102 (2009).
  * Chen  _et al._ (2010) C. H. K. Chen, T. S. Horbury, A. A. Schekochihin, R. T. Wicks, O. Alexandrova, and J. Mitchell, Anisotropy of Solar Wind Turbulence between Ion and Electron Scales, Physical Review Letters 104, 255002 (2010), arXiv:1002.2539 [physics.space-ph] .
  * Chen  _et al._ (2013) C. H. K. Chen, S. Boldyrev, Q. Xia, and J. C. Perez, Nature of Subproton Scale Turbulence in the Solar Wind, Physical Review Letters 110, 225002 (2013), arXiv:1305.2950 [physics.space-ph] .
  * Matteini  _et al._ (2012) L. Matteini, P. Hellinger, S. Landi, P. M. Trávníček, and M. Velli, Ion Kinetics in the Solar Wind: Coupling Global Expansion to Local Microphysics, Space Science Reviews 172, 373 (2012).
  * Zenitani  _et al._ (2011) S. Zenitani, M. Hesse, A. Klimas, and M. Kuznetsova, New Measure of the Dissipation Region in Collisionless Magnetic Reconnection, Physical Review Letters 106, 195003 (2011), arXiv:1104.3846 [astro-ph.SR] .
  * Wan  _et al._ (2015) M. Wan, W. H. Matthaeus, V. Roytershteyn, H. Karimabadi, T. Parashar, P. Wu, and M. Shay, Intermittent Dissipation and Heating in 3D Kinetic Plasma Turbulence, Physical Review Letters 114, 175002 (2015).
  * Bandyopadhyay  _et al._ (2020) R. Bandyopadhyay, W. H. Matthaeus, T. N. Parashar, Y. Yang, A. Chasapis, B. L. Giles, D. J. Gershman, C. J. Pollock, C. T. Russell, R. J. Strangeway, R. B. Torbert, T. E. Moore, and J. L. Burch, Statistics of Kinetic Dissipation in the Earth’s Magnetosheath: MMS Observations, Physical Review Letters 124, 255101 (2020), arXiv:2005.09232 [physics.plasm-ph] .
  * Chen  _et al._ (2019) C. H. K. Chen, K. G. Klein, and G. G. Howes, Evidence for electron Landau damping in space plasma turbulence, Nature Communications 10, 740 (2019), arXiv:1902.05785 [physics.space-ph] .
  * Verscharen  _et al._ (2019) D. Verscharen, K. G. Klein, and B. A. Maruca, The multi-scale nature of the solar wind, Living Reviews in Solar Physics 16, 5 (2019), arXiv:1902.03448 [physics.space-ph] .
  * Matthaeus  _et al._ (2014) W. H. Matthaeus, S. Oughton, K. T. Osman, S. Servidio, M. Wan, S. P. Gary, M. A. Shay, F. Valentini, V. Roytershteyn, H. Karimabadi, and S. C. Chapman, Nonlinear and Linear Timescales near Kinetic Scales in Solar Wind Turbulence, Astrophysical Journal 790, 155 (2014), arXiv:1404.6569 [physics.space-ph] .
  * Sekimoto (2010) K. Sekimoto,  _Stochastic Energetics_, Vol. 799 (2010).
  * Faranda  _et al._ (2017) D. Faranda, Y. Sato, B. Saint-Michel, C. Wiertel, V. Padilla, B. Dubrulle, and F. Daviaud, Stochastic Chaos in a Turbulent Swirling Flow, Phys. Rev. Lett. 119, 014502 (2017), arXiv:1607.08409 [physics.flu-dyn] .
  * Dubrulle (2019) B. Dubrulle, Beyond Kolmogorov cascades, Journal of Fluid Mechanics 867, P1 (2019).
  * Alberti  _et al._ (2023) T. Alberti, F. Daviaud, R. V. Donner, B. Dubrulle, D. Faranda, and V. Lucarini, Chameleon attractors in turbulent flows, Chaos Solitons and Fractals 168, 113195 (2023).
  * Friedrich and Peinke (1997) R. Friedrich and J. Peinke, Description of a Turbulent Cascade by a Fokker-Planck Equation, Physical Review Letters 78, 863 (1997).
  * Davoudi and Rahimi Tabar (1999) J. Davoudi and M. R. Rahimi Tabar, Theoretical Model for the Kramers-Moyal Description of Turbulence Cascades, Physical Review Letters 82, 1680 (1999), arXiv:cond-mat/9907063 [cond-mat] .
  * Renner  _et al._ (2001a) C. Renner, J. Peinke, and R. Friedrich, Evidence of markov properties of high frequency exchange rate data, Physica A: Statistical Mechanics and its Applications 298, 499 (2001a).
  * Milan  _et al._ (2013) P. Milan, M. Wächter, and J. Peinke, Turbulent Character of Wind Energy, Physical Review Letters 110, 138701 (2013).
  * Nickelsen and Engel (2013) D. Nickelsen and A. Engel, Probing Small-Scale Intermittency with a Fluctuation Theorem, Physical Review Letters 110, 214501 (2013), arXiv:1212.3476 [cond-mat.stat-mech] .
  * Fuchs  _et al._ (2022) A. Fuchs, C. Herbert, J. Rolland, M. Wächter, F. Bouchet, and J. Peinke, Instantons and the Path to Intermittency in Turbulent Flows, Physical Review Letters 129, 034502 (2022), arXiv:2106.08790 [physics.flu-dyn] .
  * Strumik and Macek (2008) M. Strumik and W. M. Macek, Testing for markovian character and modeling of intermittency in solar wind turbulence, Physical Review E 78, 026414 (2008).
  * Benella  _et al._ (2022a) S. Benella, M. Stumpo, G. Consolini, T. Alberti, V. Carbone, and M. Laurenza, Markovian Features of the Solar Wind at Subproton Scales, The Astrophysical Journal Letters 928, L21 (2022a), arXiv:2201.04864 [physics.space-ph] .
  * Benella  _et al._ (2022b) S. Benella, M. Stumpo, G. Consolini, T. Alberti, M. Laurenza, and E. Yordanova, Kramers-Moyal analysis of interplanetary magnetic field fluctuations at sub-ion scales, Rendiconti Lincei. Scienze Fisiche e Naturali 10.1007/s12210-022-01108-2 (2022b), arXiv:2206.11498 [physics.space-ph] .
  * Macek  _et al._ (2023) W. M. Macek, D. Wójcik, and J. L. Burch, Magnetospheric multiscale observations of markov turbulence on kinetic scales, The Astrophysical Journal 943, 152 (2023).
  * Kiyani  _et al._ (2009) K. Kiyani, S. Chapman, Y. V. Khotyaintsev, M. Dunlop, and F. Sahraoui, Global scale-invariant dissipation in collisionless plasma turbulence, Physical Review Letters 103, 075006 (2009).
  * Osman  _et al._ (2015) K. T. Osman, K. H. Kiyani, W. H. Matthaeus, B. Hnat, S. C. Chapman, and Y. V. Khotyaintsev, Multi-spacecraft Measurement of Turbulence within a Magnetic Reconnection Jet, The Astrophysical Journal Letters 815, L24 (2015), arXiv:1508.04179 [physics.space-ph] .
  * Frisch (1995) U. Frisch,  _Turbulence. The legacy of A.N. Kolmogorov_ (1995).
  * Hardy (1916) G. H. Hardy, Weierstrass’s non-differentiable function, Transactions of the American Mathematical Society 17, 301 (1916).
  * Landau and Lifshitz (2013) L. D. Landau and E. M. Lifshitz,  _Fluid Mechanics: Course of Theoretical Physics_ , Vol. 6 (Elsevier, 2013).
  * Flandrin (1989) P. Flandrin, On the spectrum of fractional brownian motions, IEEE Transactions on information theory 35, 197 (1989).
  * Benzi  _et al._ (1993) R. Benzi, S. Ciliberto, R. Tripiccione, C. Baudet, F. Massaioli, and S. Succi, Extended self-similarity in turbulent flows, Physical Review E 48, R29 (1993).
  * Renner  _et al._ (2001b) C. Renner, J. Peinke, and R. Friedrich, Experimental indications for markov properties of small-scale turbulence, Journal of Fluid Mechanics 433, 383 (2001b).
  * Arnèodo  _et al._ (2008) A. Arnèodo, R. Benzi, J. Berg, L. Biferale, E. Bodenschatz, A. Busse, E. Calzavarini, B. Castaing, M. Cencini, L. Chevillard, R. T. Fisher, R. Grauer, H. Homann, D. Lamb, A. S. Lanotte, E. Lévèque, B. Lüthi, J. Mann, N. Mordant, W. C. Müller, S. Ott, N. T. Ouellette, J. F. Pinton, S. B. Pope, S. G. Roux, F. Toschi, H. Xu, and P. K. Yeung, Universal Intermittent Properties of Particle Trajectories in Highly Turbulent Flows, Physical Review Letters 100, 254504 (2008), arXiv:0802.3776 [nlin.CD] .
  * Reinke  _et al._ (2018) N. Reinke, A. Fuchs, D. Nickelsen, and J. Peinke, On universal features of the turbulent cascade in terms of non-equilibrium thermodynamics, Journal of Fluid Mechanics 848, 117 (2018), arXiv:1702.03679 [physics.flu-dyn] .
  * Risken (1996) H. Risken,  _The Fokker-Planck Equation_ (Springer, 1996) pp. 63–95.
  * Chhiber  _et al._ (2021) R. Chhiber, W. H. Matthaeus, T. A. Bowen, and S. D. Bale, Subproton-scale intermittency in near-sun solar wind turbulence observed by the parker solar probe, The Astrophysical Journal Letters 911, L7 (2021).
  * Alberti  _et al._ (2019) T. Alberti, G. Consolini, V. Carbone, E. Yordanova, M. Marcucci, and P. De Michelis, Multifractal and Chaotic Properties of Solar Wind at MHD and Kinetic Domains: An Empirical Mode Decomposition Approach, Entropy 21, 320 (2019).
  * Pezzi  _et al._ (2019) O. Pezzi, D. Perrone, S. Servidio, F. Valentini, L. Sorriso-Valvo, and P. Veltri, Proton-Proton Collisions in the Turbulent Solar Wind: Hybrid Boltzmann-Maxwell Simulations, The Astrophysical Journal 887, 208 (2019), arXiv:1903.03398 [physics.plasm-ph] .
  * Pezzi  _et al._ (2021) O. Pezzi, H. Liang, J. L. Juno, P. A. Cassak, C. L. Vásconez, L. Sorriso-Valvo, D. Perrone, S. Servidio, V. Roytershteyn, J. M. TenBarge, and W. H. Matthaeus, Dissipation measures in weakly collisional plasmas, Monthly Notices of the Royal Astronomical Society 505, 4857 (2021), arXiv:2101.00722 [physics.plasm-ph] .
  * Servidio  _et al._ (2015) S. Servidio, F. Valentini, D. Perrone, A. Greco, F. Califano, W. H. Matthaeus, and P. Veltri, A kinetic model of plasma turbulence, Journal of Plasma Physics 81, 325810107 (2015).
  * Cerri  _et al._ (2017) S. S. Cerri, S. Servidio, and F. Califano, Kinetic cascade in solar-wind turbulence: 3d3v hybrid-kinetic simulations with electron inertia, The Astrophysical Journal 846, L18 (2017).
  * Franci  _et al._ (2018) L. Franci, S. Landi, A. Verdini, L. Matteini, and P. Hellinger, Solar wind turbulent cascade from MHD to sub-ion scales: Large-size 3d hybrid particle-in-cell simulations, The Astrophysical Journal 853, 26 (2018).
  * Leonardis  _et al._ (2016) E. Leonardis, L. Sorriso-Valvo, F. Valentini, S. Servidio, F. Carbone, and P. Veltri, Multifractal scaling and intermittency in hybrid Vlasov-Maxwell simulations of plasma turbulence, Physics of Plasmas 23, 022307 (2016).
  * Narita  _et al._ (2013) Y. Narita, K. H. Glassmeier, U. Motschmann, and M. Wilczek, Doppler shift and broadening in solar wind turbulence, Earth, Planets and Space 65, e5 (2013).
  * Chhiber  _et al._ (2019) R. Chhiber, A. V. Usmanov, W. H. Matthaeus, T. N. Parashar, and M. L. Goldstein, Contextual Predictions for Parker Solar Probe. II. Turbulence Properties and Taylor Hypothesis, The Astrophysical Journal Supplement Series 242, 12 (2019), arXiv:1902.03340 [astro-ph.SR] .
  * Shebalin  _et al._ (1983) J. V. Shebalin, W. H. Matthaeus, and D. Montgomery, Anisotropy in mhd turbulence due to a mean magnetic field, Journal of Plasma Physics 29, 525–547 (1983).
  * Matthaeus  _et al._ (1996) W. H. Matthaeus, S. Ghosh, S. Oughton, and D. A. Roberts, Anisotropic three-dimensional mhd turbulence, Journal of Geophysical Research: Space Physics 101, 7619 (1996).
  * Verma (2004) M. K. Verma, Statistical theory of magnetohydrodynamic turbulence: recent results, Physics Reports 401, 229 (2004), arXiv:nlin/0404043 [nlin.CD] .
  * Matthaeus  _et al._ (2012) W. H. Matthaeus, S. Servidio, P. Dmitruk, V. Carbone, S. Oughton, M. Wan, and K. T. Osman, Local Anisotropy, Higher Order Statistics, and Turbulence Spectra, The Astrophysical Journal 750, 103 (2012), arXiv:1201.4127 [physics.space-ph] .
  * Sahraoui  _et al._ (2013) F. Sahraoui, S. Y. Huang, G. Belmont, M. L. Goldstein, A. Rétino, P. Robert, and J. De Patoul, Scaling of the Electron Dissipation Range of Solar Wind Turbulence, The Astrophysical Journal 777, 15 (2013), arXiv:1303.7394 [astro-ph.SR] .
  * Carbone  _et al._ (2022) V. Carbone, D. Telloni, F. Lepreti, and A. Vecchio, High-frequency Magnetic Fluctuations in Space Plasmas and the Role of Electron Landau Damping, The Astrophysical Journal Letters 924, L26 (2022).
  * Nguyen  _et al._ (2020) F. Nguyen, J.-P. Laval, and B. Dubrulle, Characterizing most irregular small-scale structures in turbulence using local hölder exponents, Phys. Rev. E 102, 063105 (2020).
  * Gardiner  _et al._ (1985) C. W. Gardiner _et al._ ,  _Handbook of stochastic methods_, Vol. 3 (springer Berlin, 1985).
  * van Kampen (1992) N. G. van Kampen,  _Stochastic Processes in Physics and Chemistry_ (1992).