# Zero-shot Temporal Relation Extraction with ChatGPT

Chenhan Yuan, Qianqian Xie, Sophia Ananiadou
Department of Computer Science, The University of Manchester
{chenhan.yuan, qianqian.xie<EMAIL_ADDRESS>
###### Abstract

The goal of temporal relation extraction is to infer the temporal relation
between two events in the document. Supervised models are dominant in this
task. In this work, we investigate ChatGPT’s ability on zero-shot temporal
relation extraction. We designed three different prompt techniques to break
down the task and evaluate ChatGPT. Our experiments show that ChatGPT’s
performance has a large gap with that of supervised methods and can heavily
rely on the design of prompts. We further demonstrate that ChatGPT can infer
more small relation classes correctly than supervised methods. The current
shortcomings of ChatGPT on temporal relation extraction are also discussed in
this paper. We found that ChatGPT cannot keep consistency during temporal
inference and it fails in actively long-dependency temporal inference.

##  1 Introduction

The temporal relation extraction task aims to extract the temporal relation
between either two event triggers in the given document Dligach et al.
(2017a). In this way, a timeline of events in the document can be constructed.
It is a crucial task for many downstream NLP tasks, such as natural language
understanding Mani et al. (2006); Paul et al. (2017), storyline construction
Do et al. (2012); Minard et al. (2015), and temporal question answering Jia et
al. (2018b, a), etc. Conventionally, recent temporal relation extraction (RE)
models are fine-tuned based on pre-trained language models (PLMs), such as
BERT and RoBERTa Devlin et al. (2019); Liu et al. (2019). On top of the PLMs,
complex neural networks are applied to classify the temporal relations, such
as self-attention Lin et al. (2019); Ning et al. (2018a), graph convolutional
networks (GCNs) Mathur et al. (2021), and policy network Man et al. (2022).
Most well-performed temporal relation extractors are supervised models, that
is, they heavily rely on annotated training documents first before extracting
temporal relations on the testing set. However, annotating the temporal
relations in training documents requires much domain experts’ efforts Naik et
al. (2019); Ning et al. (2018b), which is a high cost.

Different from supervised learning methods, zero-shot learning (ZSL) Xian et
al. (2017) aims to train the model that can be generalized to unseen data
without annotated training data and has attracted much attention in recent
years. Most recently, large language models (LLMs) Brown et al. (2020); Bubeck
et al. (2023) such as ChatGPT111https://openai.com/blog/chatgpt have exhibited
remarkable ability in zero-shot learning on various natural language
processing (NLP) and medical tasks Bang et al. (2023), such as information
extraction Wei et al. (2023), machine translation Jiao et al. (2023),
summarization evaluation Luo et al. (2023), and mental health detection Yang
et al. (2023). However, the performance of LLMs in detecting temporal
relations between events are not explored yet. Therefore, it is an urgent and
spontaneous question if the LLM can perform zero-shot temporal relation
extraction tasks well given a proper prompt approach, and if LLM can be the
new paradigm for temporal RE.

In this paper, we explore the performance of ChatGPT on the zero-shot temporal
relation extraction, and propose three different prompt strategies to interact
with ChatGPT. More specifically, we start with the simple zero-shot prompt
that directly requires ChatGPT to infer the temporal relation given the
document. Then, we design the event ranking prompt, where ChatGPT is asked to
infer the events shown in the given document instead of inferring temporal
relations. Finally, we propose the chain-of-thought (CoT) prompt Wei et al.
(2022) to break down the task into two-stage, which guides ChatGPT to make
temporal relation reasoning step by step. Based on our experimental results
and analysis, we have the following findings:

  * •

Overall Performance. ChatGPT significantly underperforms advanced supervised
methods and even traditional neural network methods such as Bi-LSTM,
indicating the challenge of temporal relation detection with ChatGPT without
task-specific fine-tuning.

  * •

Prompts. Similar to the finding of recent efforts, the CoT prompt can
significantly improve ChatGPT’s performance compared with other prompts across
all datasets, indicating the importance of proper prompt engineering.

  * •

Limitations. Compared with supervised methods, ChatGPT has better performance
in the temporal relations with small proportions in the dataset. However, it
is also found to have limitations in detecting long-dependency temporal
relation extraction and inconsistent temporal relation inference.

##  2 Related Work

###  2.1 Temporal Relation Extraction

Several studies have explored the use of temporal information in relation
extraction. For example, Han et al. Han et al. (2019b, a) incorporated tense
information and timex temporal interactions into their models. Other
researchers have proposed using graph neural networks (GNNs) to encode
dependency structures, which are important for temporal relation extraction
Mathur et al. (2021); Schlichtkrull et al. (2018). Wang et al. Wang et al.
(2022b) added an attention layer to an R-GCN-based model to focus on document-
creation-time (DCT). Man et al. Man et al. (2022) used a reinforcement
learning framework to select optimized sentences for input into neural models,
improving performance.

In the clinical domain, Leeuwenberg et al. Leeuwenberg and Moens (2017)
applied integer linear programming constraints to learn structured temporal
relations. Dligach et al. Dligach et al. (2017b) improved performance by using
neural networks such as CNN and LSTM as the backbone model. Lin et al. Lin et
al. (2019) utilized pre-trained language models like BERT to learn
contextualized embeddings. However, no work has yet explored the feasibility
of using LLMs in temporal relation extraction.

###  2.2 Zero-shot Learning with ChatGPT

Since it was launched, ChatGPT has drawn much attention to its strong ability
for various NLP tasks. In the clinical domain, Tang et al. explored ChatGPT’s
ability on zero-shot named entity recognition and relation extraction Tang et
al. (2023). The experiments on NCBI Disease and BC5CDR Chemical datasets
showed that ChatGPT cannot recognize named entities correctly in the clinical
domain as the F1 drops around 55.94%-91.58% compared to SOTA-supervised
methods. ChatGPT’s poor clinical NER was also proved by testing in i2b2
dataset Hu et al. (2023). However, it can achieve comparable performance in
clinical relation extraction as the F1 score only decreased by 4.73%-10.93%
Tang et al. (2023); Agrawal et al. (2022).

In extraction-related tasks, some work evaluated ChatGPT’s ability in event
extraction, general information extraction, and relation extraction Tang et
al. (2023); Gao et al. (2023); Wei et al. (2023). The evaluation process
follows multi-stage interactions/conversations with ChatGPT and guides it to
produce the desired answers. The evaluation results of these studies showed
that with proper prompting, ChatGPT can achieve comparable performance with
the supervised methods on zero-shot or few-shot settings of extraction tasks.
However, there is also some work pointing out that ChatGPT’s ability is still
limited in some specific extraction scenarios such as extracting clinical
notes with privacy information masked Tang et al. (2023). Some work also
discussed that for non-digital-available texts, such as historical documents,
the entity recognition was performed poorly by ChatGPT González-Gallardo et
al. (2023); Borji (2023).

##  3 ChatGPT for Zero-shot Temporal Relation Extraction

Figure 1: The proposed prompts.

Given the input document, the temporal relation extraction aims to identify
the temporal relation between any two event triggers in the document, which is
modeled as the multi-classification problem. We propose three different
prompts methods to evaluate ChatGPT’s performance on temporal relation
extraction as shown in Figure 1.

###  3.1 Zero-shot Prompt

In this prompt, given the document $D=\\{x_{1},x_{2},x_{3},\cdots,x_{n}\\}$,
we first label the event triggers with <EVENT></EVENT>. That is if $x_{i}$ is
an event trigger, we label it as <EVENT>$x_{i}$</EVENT>. Then the whole
labeled document is sent to ChatGPT and we ask ChatGPT to find the temporal
relations between either two events. Note that our goal is to test ChatGPT’s
zero-shot ability in the temporal relation extraction task. Therefore, we do
not provide any examples in our prompts. As shown in Figure. 1, we give
ChatGPT the whole document, the list of all temporal relations, and the
annotations of event triggers. In the end, the input question is designed as
“what is the temporal relation between <EVENT>$x_{i}$</EVENT> and
<EVENT>$x_{j}$</EVENT>?”. We let ChatGPT to answer the question by using the
temporal relations provided in the list.

###  3.2 Event Ranking Prompt

Further, we design a new prompt to make the task easier to learn for ChatGPT.
Specifically, given the document $D$ and one event trigger $e_{i}$ with the
form <EVENT>$x_{i}$</EVENT> and temporal relation set $R$, we require ChatGPT
to complete the query $(e_{i},r_{j},?)\>\forall r_{j}\in R$. In this way,
instead of querying $(e_{i},?,e_{j})$ as in the previous prompt, ChatGPT is
required to predict the missing event trigger. As event triggers are already
shown in the given text/document, ChatGPT is more likely to infer the event
triggers than unseen temporal relations as they are not explicitly provided in
the context. In detail, as shown in Fig. 1, we achieve this by asking the
question first, such as “Which events happened before $e_{i}$?”. Then based on
the feedback of ChatGPT, we form the $<e_{i},r,e_{j}>$ triplets to perform the
evaluation. Note that if the same event pair is detected in different temporal
relation (>2), we denote that this event pair has a “vague” relation as
ChatGPT cannot confidently determine which temporal relation the event pair is
classified. We also asked ChatGPT to provide some concise prompts to perform
temporal relation extraction tasks. As shown in Fig 2, the prompts provided by
ChatGPT are in line with our event ranking prompt approach.

Figure 2: Prompts generated by ChatGPT.

###  3.3 Chain-of-thought Prompt

We notice that if two event triggers refer to the same event but point to
different timestamps of the duration of that event, ChatGPT cannot distinguish
them. For example, in Figure. 1, “<EVENT e3> started” and “<EVENT e6> ended”
referred to the beginning and the end of the season event. ChatGPT assumes
that these two event triggers happened at the same time if we directly ask
about the temporal relation following the zero-shot prompt. Therefore, we
propose a new chain-of-thought prompt with two steps, which firstly navigates
ChatGPT to distinguish event triggers referring to the same event, and then
guides ChatGPT to infer their temporal relation. Specifically, given the
document $D$ and two event triggers $e_{1}$ and $e_{2}$, we first ask ChatGPT
to determine if $e_{1}$ and $e_{2}$ refer to the same event. If they are not,
we further ask ChatGPT to determine the temporal relation between the two
event triggers. If they point to the same event, we use a similar prompt but
with the extra phrase “in that event” to ask ChatGPT. As shown in Fig. 1, we
first ask ChatGPT if the two event triggers <EVENT>started</EVENT> and
<EVENT>ended</EVENT> refer to the same event. Then based on ChatGPT’s
feedback, we further iteratively go through the whole temporal relation list
to determine which temporal relation exists between the two event triggers. We
show the full prompts we designed in detail in Appendix A.

##  4 Experiments

###  4.1 Datasets

We use three datasets to evaluate the performance of ChatGPT on the zero-shot
temporal relation extraction. The statistical details of these datasets are
shown in the Table 1.

Dataset | Train | Dev | Test | Labels
---|---|---|---|---
TB-Dense | 4,032 | 629 | 1,427 | 6
MATRES | 6,336 | – | 837 | 4
TDDMan | 4,000 | 650 | 1,500 | 5
Table 1: Statistics of the number of annotated event pairs and different
temporal relation labels of the MATRES, TB-Dense and TDDMan datasets.

  * •

TimeBank-Dense Cassidy et al. (2014) labeled 36 news documents in total. The
temporal relations between event triggers-event triggers, timex-event
triggers, timex-timex, are labeled. Following previous work, we only test our
model on the event trigger-event trigger relation in the testing set.

  * •

MATRES Ning et al. (2018b) is a dataset primarily focusing on temporal
relations of event triggers with local sentences (1 or 2 sentences).

  * •

TDDiscourse Naik et al. (2019) was created to explicitly emphasize global
discourse-level temporal ordering. Based on the annotation accuracy, the
dataset is split into TDDMan and TDDAuto, where TDDAuto introduces much more
automatic labels and noise. In this paper, we only evaluate ChatGPT on TDDMan
because of the budget limitation.

We only use the testing set of each dataset directly to test our approaches as
we do not require training of ChatGPT, following the zero-shot setting. We
report the F1 score on each dataset and each temporal relation.

###  4.2 Baseline Models

Since there is no zero-shot learning methods for temporal relation extraction
before, we compare the performance of ChatGPT with the following advanced
supervised methods:

  * •

CAEVO Chambers et al. (2014) a sieve-based architecture that includes multi-
level classifiers for intra-sentence temporal relation learning.

  * •

SP+ILP Ning et al. (2017) a structured learning approach that captures the
global temporal features when inferring the relation between two local events.

  * •

Bi-LSTM Cheng and Miyao (2017) a Bi-LSTM-based model that encodes the
dependency path between two events to classify temporal relation.

  * •

Joint Han et al. (2019b) a joint end-to-end event and temporal relation
extraction model that shares the contextualized embedding of two sub-tasks

  * •

Deep Han et al. (2019a) a neural network model that utilized SSVM as the
scoring function to learn temporal constraints and context embeddings.

  * •

UCGraph Liu et al. (2021) a graph-based model that is trained with mask pre-
training mechanism. The model’s uncertainty level is used to guide the
inference during testing.

  * •

TIMERS Mathur et al. (2021) a graph-based model that leverages three graphs to
learn temporal, rhetorical, and syntactic information, respectively.

  * •

SCS-EERE Man et al. (2022) a reinforcement learning-based selector is designed
to select the optimized sentences for temporal inference between the given two
events.

  * •

RSGT Zhou et al. (2022) a syntactic-and-semantic-based graph model pre-trained
on a temporal neighbor prediction task.

  * •

FaithTRE Wang et al. (2022a) a model that applied Dirichlet prior to
estimating the correctness likelihood. A temperature scaling is also used to
recalibrate the model confidence measure after bias mitigation.

  * •

DTRE Wang et al. (2022b) a document creation time(DCT)-aware graph with a
global consistency mechanism when inferring temporal relations.

  * •

MulCo Yao et al. (2022) a joint model using the BERT to learn contextualized
features and GNN to capture syntactic structures. The two models are combined
via a multi-level contrastive learning framework.

###  4.3 Results

Models | MATRES | TDDMan | TB-Dense
---|---|---|---
prec | recall | F1 | prec | recall | F1 | prec | recall | F1
CAEVO Chambers et al. (2014) | – | – | – | 32.3 | 10.7 | 16.1 | 49.9 | 46.6 | 48.2
SP+ILP Ning et al. (2017) | 71.3 | 82.1 | 76.3 | 23.9 | 23.8 | 23.8 | 58.4 | 58.4 | 58.4
Bi-LSTM Cheng and Miyao (2017) | 59.5 | 59.5 | 59.5 | 24.9 | 23.8 | 24.3 | 63.9 | 38.9 | 48.4
Joint Han et al. (2019b) | – | – | 75.5 | 41.0 | 41.1 | 41.1 | – | – | 64.5
Deep Han et al. (2019a) | 77.4 | 86.4 | 81.7 | – | – | – | 62.7 | 58.9 | 62.5
UCGraph Liu et al. (2021) | – | – | – | 44.5 | 42.3 | 43.4 | 62.4 | 56.1 | 59.1
TIMERS Mathur et al. (2021) | 81.1 | 84.6 | 82.3 | 43.7 | 46.7 | 45.5 | 48.1 | 65.2 | 67.8
SCS-EERE Man et al. (2022) | 78.8 | 88.5 | 83.4 | – | – | 51.1 | – | – | –
FaithTRE Wang et al. (2022a) | – | – | 82.7 | – | – | 52.9 | – | – | –
RSGT Zhou et al. (2022) | 82.2 | 85.8 | 84.0 | – | – | – | 68.7 | 68.7 | 68.7
DTRE Wang et al. (2022b) | – | – | – | 56.3 | 56.3 | 56.3 | – | – | 70.2
MulCo Yao et al. (2022) | 88.2 | 88.2 | 88.2 | 56.2 | 54.0 | 55.1 | 84.9 | 84.9 | 84.9
ChatGPT_ZS | 26.4 | 24.3 | 25.3 | 17.7 | 13.6 | 15.3 | 23.7 | 14.3 | 17.8
ChatGPT_ER | 21.9 | 17.3 | 19.3 | 3.7 | 0.3 | 0.5 | 37.6 | 35.8 | 36.6
ChatGPT_CoT | 48.0 | 57.7 | 52.4 | 26.8 | 22.3 | 24.3 | 43.4 | 32.2 | 37.0
Table 2: The comparison of ChatpGPT with various prompt techniques and
supervised state-of-the-art models.

In table 2, we can see that ChatGPT struggles to outperform supervised state-
of-the-art models such as SCS-EERE Man et al. (2022) and RSGT Zhou et al.
(2022), and even the traditional neural networks methods such as CAEVO and BI-
LSTM, indicating its ineffective in the temporal relation extraction task in
the zero-shot setting. Table 2 also shows the performance of ChatGPT under
different prompts in three datasets. We noticed that ChatGPT_ER yields the
worst performance on the MATRES and TDDMan datasets. ChatGPT’s performance
with the event ranking prompt on TDD-Man dataset is poor as most event trigger
pairs cannot be detected under this prompt. However, if the event trigger
pairs are explicitly fed to ChatGPT, it can somehow partially infer the
temporal relations correctly. For example, the zero-shot prompt and CoT prompt
could improve the F1 score by $14.8\%$ and $23.8\%$, respectively. While for
the TB-Dense dataset, ChatGPT_ZS has the worst performance. ChatGPT_CoT
achieves the best performance across all datasets, such as it significantly
outperforms ChatGPT_ZS and ChatGPT_ER by 27.1% and 33.1%. This illustrates the
effectiveness of the CoT prompt with step-by-step guidance in prompting
ChatGPT.

Relation | Zero-shot | CoT | Event ranking | Deep
---|---|---|---|---
prec | recall | F1 | prec | recall | F1 | prec | recall | F1 | prec | recall | F1
overall | 26.4 | 24.3 | 25.3 | 48.0 | 57.7 | 52.4 | 21.9 | 17.3 | 19.3 | 77.4 | 86.4 | 81.7
EQUAL | 0.0 | 0.0 | 0.0 | 7.1 | 2.9 | 4.1 | 5.8 | 11.1 | 7.6 | 0.0 | 0.0 | 0.0
VAGUE | 14.3 | 58.7 | 23.1 | 14.4 | 8.1 | 10.4 | 14.6 | 86.2 | 25.0 | 0.0 | 0.0 | 0.0
AFTER | 34.0 | 25.6 | 29.2 | 41.6 | 41.8 | 41.7 | 36.4 | 1.6 | 3.0 | 72.3 | 84.8 | 78.0
BEFORE | 52.5 | 17.8 | 26.6 | 63.1 | 71.6 | 67.1 | 57.0 | 13.0 | 21.1 | 80.1 | 89.6 | 84.6
Table 3: The zero-shot performance of ChatGPT with three different prompts on the MATRES dataset. Relation | Zero-shot | CoT | Event ranking
---|---|---|---
prec | recall | F1 | prec | recall | F1 | prec | recall | F1
overall | 17.7 | 13.6 | 15.3 | 26.8 | 22.3 | 24.3 | 3.7 | 0.3 | 0.5
is included | 9.5 | 0.7 | 1.3 | 20.9 | 3.1 | 5.4 | 0.0 | 0.0 | 0.0
include | 41.9 | 17.7 | 24.8 | 37.9 | 11.2 | 17.3 | 0.0 | 0.0 | 0.0
after | 14.7 | 9.0 | 11.2 | 33.3 | 4.3 | 7.5 | 0.0 | 0.0 | 0.0
before | 29.7 | 22.9 | 25.9 | 35.1 | 70.8 | 46.9 | 12.5 | 0.7 | 1.4
simultaneous | 3.9 | 39.1 | 7.0 | 0.0 | 0.0 | 0.0 | 11.1 | 2.2 | 3.6
Table 4: The zero-shot performance of ChatGPT with three prompts on the TDD-
Man dataset.

Table 3, Table 4 and Table 5 further list the detail results of ChatGPT with
different prompts on three datasets. We can see that the performance with the
event ranking prompt is much better than that on other datasets. ChatGPT with
the zero-shot prompt cannot determine the temporal relation “is included” and
therefore yields a 0.0 performance in this type of relation. The CoT prompt
improves the overall performance by significantly detecting “before” and
“after” temporal relations. As these two relations take a great portion of the
whole dataset, the overall performance is also improved.

Relation | Zero-shot | CoT | Event ranking
---|---|---|---
prec | recall | F1 | prec | recall | F1 | prec | recall | F1
overall | 23.7 | 14.3 | 17.8 | 43.4 | 32.2 | 37.0 | 37.6 | 35.8 | 36.6
is included | 0.0 | 0.0 | 0.0 | 10.0 | 1.9 | 3.2 | 6.2 | 3.8 | 4.7
include | 3.3 | 10.7 | 24.8 | 5.5 | 16.1 | 8.2 | 16.7 | 5.4 | 8.1
after | 29.0 | 17.2 | 11.2 | 70.4 | 13.9 | 23.2 | 19.0 | 1.5 | 2.7
before | 40.0 | 9.9 | 25.9 | 35.0 | 75.5 | 47.9 | 31.2 | 25.3 | 27.9
simultaneous | 1.5 | 45.5 | 3.0 | 33.3 | 4.5 | 8.0 | 6.7 | 50.0 | 11.8
vague | 44.6 | 24.0 | 31.2 | 51.2 | 29.6 | 37.5 | 46.0 | 63.5 | 53.4
Table 5: The zero-shot performance of ChatGPT with three prompts on the
TimeBank-Dense dataset.

##  5 Discussion

###  5.1 ChatGPT is slightly better on small temporal relation classes

The imbalanced data is a severe long-existing problem in the temporal relation
extraction task. Because of the events’ temporal order frequency in real life,
some temporal relations, such as “simultaneous” and “equal”, are very limited
in most temporal relation extraction datasets. And popular NLP data-augmented
methods are difficult to be applied in the temporal domain. Therefore, most
state-of-the-art supervised methods yield much worse performance on small
relation classes.

In Deep Han et al. (2019a), the authors reported detailed performance on each
temporal class in MATRES dataset. We therefore especially compare the
performance on small relation classes against Deep, i.e., “EQUAL” and “VAGUE”.
As shown in Table 3, the supervised model Deep could achieve much better
overall performance ($81.7$ F1 score), due to the contribution of two majority
relations, “before” and “after”. Compared to Deep’s 0.0 performance on “EQUAL”
and “VAGUE”, ChatGPT with event ranking, CoT and zero-shot prompts can
correctly extract some small class relations.

###  5.2 ChatGPT failed in actively long-dependency temporal relation
extraction

As shown in Table 2, ChatGPT’s performance drops a lot in the TDDMan dataset.
We argue that this is mainly because the TDDMan focuses more on discourse-
level temporal relations and ChatGPT failed to extract useful information from
long documents. As shown in Table 4, the event ranking prompt yields almost
0.0 on the whole dataset. In practice, we initially input the whole document
$D$ to ChatGPT and ask ChatGPT which event triggers in the document $D$
happened before the given event trigger $e_{1}$. That is, ChatGPT should
actively search all event triggers in the document and produce answers.
However, in the TDDMan dataset, ChatGPT cannot produce a formatted answer and
the outputs sometimes are even some random words in the document instead of
event triggers. We then test the limit of the size of the input document,
i.e., the number of sentences. We finally found that if we limit the size of
the input document to at most 8 context sentences around the event triggers,
ChatGPT would be more stable to produce formatted answers instead of repeating
part of the document randomly. However, in this way, most temporal relations
extracted by ChatGPT do not match with the golden labels in the TDDMan dataset
because the extracted temporal relations are only in short dependency while
TDDMan emphasizes long-distance temporal dependency.

Nevertheless, surprisingly, if we explicitly ask ChatGPT what the temporal
relations between two event triggers labeled in the document are, ChatGPT can
answer some of them correctly. Note that we do not cut the size of the
document in this case and ChatGPT can still learn some of the temporal
dependency in the document. Compared to the event ranking prompt, ChatGPT
passively receives two event trigger information with the zero-shot and CoT
prompts. This may reduce the inference difficulty as more information is
given.

###  5.3 ChatGPT can be improved via multi-stage “yes” or “no” prompts

Intuitively, the most efficient way to query ChatGPT about temporal relations
between two events in the given document should be the zero-shot prompt, which
directly asks the ChatGPT to answer the temporal relation between any two
event triggers. If only one event trigger is given, then the prompts provided
by ChatGPT, i.e., event ranking, should be used to interact with ChatGPT.
However, our extensive experiments show that these two prompt methods produce
much worse performance in most cases compared to the CoT prompt. Note that
both zero-shot and CoT provide sufficient information about event triggers. We
argue there is another difference resulting in the performance gap.

Comparing the two prompts, one significant difference is that the CoT prompt
only accepts “yes” or “no” answers while the zero-shot prompt returns a
specific temporal relation label. In the zero-shot prompt, ChatGPT is required
to select a temporal relation from the given list, which is similar to a
conventional multi-class classification problem. However, in the CoT prompt,
ChatGPT only has to determine if one specific temporal relation exists (or
not) between two event triggers. This simplified the problem into a binary
classification. Further, with the previous question-answer pair as context,
ChatGPT has a higher probability of making the correct selection. For example,
in Fig 1, in the first round, ChatGPT already inferred that the relation is
not “EQUAL”. Then in the second round, ChatGPT is more confident to predict
the temporal relation from _[BEFORE, AFTER, VAGUE]_ instead of “EQUAL”.

###  5.4 ChatGPT’s temporal inference is inconsistent even with sufficient
context

Figure 3: ChatGPT’s two temporal inconsistency cases examples in the Event
ranking prompt.

During the testing of ChatGPT with the event ranking prompt, we noticed a
fatal issue in ChatGPT’s temporal relation extraction, namely the inconsistent
temporal relation inference. Given the same input document, ChatGPT may
produce different temporal relations between two event triggers. As the top
example shown in Figure 3, given the document $D$, if the prompt is “Which
event triggers happened _before_ $e_{1}$?”, ChatGPT will give a list of event
triggers, e.g, [$e_{2},e_{5},e_{6}$]. Now given the same document $D$, if the
prompt is “Which event triggers happened _after_ $e_{6}$”, ChatGPT is expected
to at least include $e_{1}$ in the list. However, during the experiments, we
noticed that ChatGPT failed in this scenario multiple times and the failures
can be categorized into two cases. The first failure case is that ChatGPT does
not include $e_{1}$ in any list associated with $e_{6}$. The second case is
that ChatGPT includes $e_{1}$ in a wrong list, e.g., “EQUAL”, associated with
$e_{6}$ which violates the temporal consistency.

Figure 4: ChatGPT’s inconsistency failures examples in the CoT prompt. The top
shows ChatGPT does not reply “vague” when unsure and the bottom shows that
ChatGPT still infers other inconsistent relations.

A similar problem also happened to the CoT prompt. As shown in Figure 4,
during the experiments, ChatGPT would give various answers instead of “vague”
as we specified, if it thinks there is no clue to infer the temporal relation
between the given two event triggers. These answers include “Cannot
determine.”, “I cannot answer that question as it is unclear from the given
information.”, “Unknown.”, etc. We treat all of these answers as “vague” when
we evaluate ChatGPT. Intuitively, if ChatGPT cannot answer “yes” or “no” for
the specific temporal relation between two event triggers, it should also not
be able to answer questions about other temporal relations of the same event
triggers. However, ChatGPT may violate its “unknown” statement in the multi-
stage prompts. We test if ChatGPT’s inconsistency exists in multi-stage
prompts by implementing the following experiments. For example, during the
$i$-th round, if we ask “Did $e_{1}$ happen before $e_{2}$?” and ChatGPT’s
answer includes “It is unclear from the given information.”. We then ask the
next $i+1$ question “Did $e_{1}$ happen after $e_{2}$?”. And surprisingly, in
most cases ($84\%$), ChatGPT can classify the event trigger pair into other
temporal relation classes even if it claims that the information is
insufficient. Moreover, $96\%$ of these classification results is incorrect.

##  6 Conclusion

In this work, we comprehensively test ChatGPT’s zero-shot ability on temporal
relation extraction. We designed three different prompts to evaluate ChatGPT’s
performance. Our extensive experiments demonstrate that with proper prompting,
ChatGPT’s performance on zero-shot temporal relation extraction can be
significantly improved, highlighting the importance of prompt engineering to
better trigger ChatGPT’s ability in future work. However, compared to
supervised methods, ChatGPT is still far behind. We further discuss our
findings from experimental results, including its better performance in small
classes than supervised methods and its drawbacks such as failures in long-
distance temporal dependency inference and inconsistent temporal relation
inference. Our work only takes the initial step of exploring the LLMs for
zero-shot temporal relation extraction. To fill the gap between the
performance of LLMs in the zero-shot setting and that of advanced supervised
methods, we believe more efforts should be explored in the future, such as the
few-shot prompt engineering and investigating the performance of other LLMs
such as GPT-4 Bubeck et al. (2023).

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##  Appendix A Detailed Prompts

In this section, we provide detailed prompts we used in the experiments.

For datasets with the following temporal relations, such as MATRES: _[before,
after, vague, equal]_ , the following prompt is used for Zero-shot method:

> Given the document $D$ and a list of temporal relations [before, after,
> vague, equal] and event triggers are labeled as [EVENT][/EVENT]. What is the
> temporal relation between $e_{1}$ and $e_{2}$? Answer vague if unsure. keep
> the answer short and concise

The prompts designed for Event ranking are shown as follows:

> Given the document $D$ and event triggers that are labeled as
> [EVENT][/EVENT], which event triggers happened “relation” $e_{1}$? keep the
> answer short and concise {If the relation that will be asked is “before” or
> “after”} Given the document $D$ and event triggers that are labeled as
> [EVENT][/EVENT], which event triggers were happening at the same time as
> $e_{1}$? keep the answer short and concise {If the relation that will be
> asked is “equal”}

We propose the following prompts for CoT method:

> Is $e_{1}$ “relation” $e_{2}$? Keep the answer short and concise. {If the
> relation that will be asked is “before” or “after”} Did $e_{1}$ and $e_{2}$
> simultaneously happen? Keep the answer short and concise. {If the relation
> that will be asked is “equal”} Is the temporal relation of $e_{1}$ and
> $e_{2}$ vague? Keep the answer short and concise. {If the relation that will
> be asked is “vague”}

Some datasets accompany by the following temporal relations, such as TB-Dense
and TDDiscouse: _[before, after, vague, include, is included, simultaneous]_.
Note that “vague” can be optional for some datasets, but we still ask ChatGPT
to produce a “vague” label if it cannot infer temporal relation in the given
two events. The following prompt is used for the Zero-shot method:

> Given the document $D$ and a list of temporal relations [before, after,
> include, is included, simultaneous] and event triggers are labeled as
> [EVENT][/EVENT]. what is the temporal relation between $e_{1}$ and $e_{2}$?
> Answer vague if unsure. keep the answer short and concise

The prompts designed for Event ranking are shown as follows:

> Given the document $D$ and event triggers that are labeled as
> [EVENT][/EVENT], which event triggers happened “relation” $e_{1}$? keep the
> answer short and concise {If the relation that will be asked is “before” or
> “after”} Given the document $D$ and event triggers that are labeled as
> [EVENT][/EVENT], which event triggers were happening at the same time as
> $e_{1}$? keep the answer short and concise {If the relation that will be
> asked is “simultaneous”} Given the document $D$ and event triggers that are
> labeled as [EVENT][/EVENT], which event triggers were included in $e_{1}$?
> keep the answer short and concise {If the relation that will be asked is
> “include”} Given the document $D$ and event triggers that are labeled as
> [EVENT][/EVENT], which event triggers that $e_{1}$ were included in? keep
> the answer short and concise {If the relation that will be asked is “is
> included”}

In the CoT method, we first provide the following prompt:

> Given the document $D$, are $e_{1}$ and $e_{2}$ referring to the same event?
> Keep the answer short and concise

Then the following prompts are fed to ChatGPT if the answer tends to “NO”:

> Did $e_{1}$ happen “relation” $e_{2}$? Keep the answer short and concise.
> {If the relation that will be asked is “before” or “after”} Did $e_{1}$ and
> $e_{2}$ simultaneously happen? Say yes or no and keep the answer short and
> concise" {If the relation that will be asked is “simultaneous”} Did $e_{2}$
> occur during the time period of $e_{1}$? Keep the answer short and concise
> {If the relation that will be asked is “include”} Did $e_{1}$ occur during
> the time period of $e_{2}$? Keep the answer short and concise {If the
> relation that will be asked is “is included”}

Note that if ChatGPT determines that the two event triggers refer to the same
event, we add a phrase “in that event” in the end of each prompt mentioned
above.


# A Probabilistic Approach for Gradient Estimates on Time-Inhomogeneous
Manifolds

Li-Juan Cheng 111School of Mathematical Sciences, Beijing Normal University,
Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing
100875, The People’s Republic of China. E-mail: chenglj@mail.bnu.edu.cn(L.J.
Cheng)

(June 12, 2013)

Abstract Gradient inequalities of the Hamilton type and the Li-Yau type for
positive solutions to the heat equation are established from a probabilistic
viewpoint, which simplifies the proofs of some results of Sun [Pacific J.
Math., 253 (2011), pp. 489–510].

Keywords: Geometric flow, heat equation, Li-Yau type inequality, Hamilton type
gradient inequality, $g_{t}$-Brownian motion

MSC(2010): 58J65, 60J60

##  1 Introduction

In this paper, we want to establish gradient estimates on time-inhomogeneous
manifolds by stochastic analysis. It is well known that the gradient
estimates, known as the differential Harnack inequalities, are powerful tools
both on geometry and stochastic analysis. For example, R. Hamilton [8]
established differential Harnack inequalities for the scalar curvature under
the Ricci flow, which is applied to the singularity analysis; Perelman [11]
successfully used a differential Harnack inequalities to consider the Poincaré
conjecture.

Under some curvature constraints, Sun [13] gave gradient inequalities for
positive solutions to the heat equation under general geometric flows.
Meanwhile, Bailesteanu-Cao-Pulemotov [4], Liu [10] independently consider
these similar problems under the Ricci flow. As announced, we want to review
these inequalities from a probabilistic viewpoint. When the metric is
independent of $t$, this point of view has been worked well for local
estimates in positive harmonic function [2] and for Li-Yau type gradient
estimates [3] on Riemannian manifolds.

Let $M$ be a $d$-dimensional differential manifold without boundary equipped
with a family of complete Riemannian metrics $(g_{t})_{t\in[0,T]}$,
$T\in(0,\infty)$, which is $C^{1}$ in $t$. For simplicity, we take the
notation: for $X,Y\in TM$,

$\displaystyle\mathcal{R}_{t}(X,Y):={\rm
Ric}_{t}(X,Y)+\partial_{t}g_{t}(X,Y),$

where ${\rm Ric}_{t}$ is the Ricci curvature tensor with respect to the metric
$g_{t}$. Suppose a smooth positive function
$u:M\times[0,T]\rightarrow\mathbb{R}$ satisfies the heat equation

$\displaystyle\frac{\partial}{\partial t}u(x,\cdot)(t)=\frac{1}{2}\Delta
u(\cdot,t)(x)$ (1.1)

on $M\times[0,T]$. This paper is devoted to the Hamilton type and Li-Yau type
gradient estimates for $u$ by constructing some suitable semimartingales. To
explain the main idea, we take the Li-Yau type gradient estimate on the
compact manifold carrying the Ricci flow for example. Here and in what
follows, the Ricci flow will mean (probabilistic convention):

$\displaystyle\frac{\partial}{\partial t}g(x,t)=-{\rm Ric}(x,t),\ \ (x,t)\in
M\times[0,T].$ (1.2)

In this case, ${\mathcal{R}}_{t}=0$. Moreover, suppose that there exists some
constant $k>0$ such that

$\displaystyle 0\leq{\rm Ric}_{t}\leq k$

holds on $M\times[0,T]$. Let $X_{t}^{T}$ be a $g_{(T-t)}$-Brownian motion (see
[1] for the construction), which is ensured to be non-explosive under the
Ricci flow (see [9]). Let

$\displaystyle\hat{S}_{t}$ $\displaystyle=h_{t}\left(\frac{|\nabla
u|^{2}}{u}(X^{T}_{T-t},t)-\Delta u_{t}\right)-nu_{t}\dot{h}_{t},\ \alpha>1,$
(1.3)

where $u_{t}:=u(X^{T}_{T-t},t)$ and $h_{t}$ is the solution of

$\dot{h}_{t}=h_{t}(c_{1}t^{-1}+c_{2}k),\ \ h_{T}=1$

for some positive constant $c_{1},c_{2}$. Then,

$\displaystyle\hat{S}_{t}=h_{t}\left\\{\frac{|\nabla
u|^{2}}{u}(X^{T}_{T-t},t)-\Delta
u(\cdot,t)(X_{T-t}^{T})-nu_{t}(c_{1}t^{-1}+c_{2}k)\right\\}.$ (1.4)

Now, the Li-Yau type gradient estimate can be derived if we can choose some
suitable constants $c_{1},c_{2}$ in (1.4) such that $\hat{S}_{t}$ is a
supermartingale, see the proof of Lemma 3.8 below. When it reduces to the
constant metric case, the classical Li-Yau inequality says that $\frac{|\nabla
u|^{2}}{u^{2}}-\frac{\Delta u}{u}$ can be dominated by $\frac{n}{t}$ and it
does not need ${\rm Ric}_{t}\leq k$. However, in our setting, when using the
Itô formula, we need this curvature condition to deal with some additional
terms from the time derivative of the metric, e.g. the derivative of $\Delta
u$ about $t$. It is the main difficult for us to overcome.

The rest parts of the paper are organized as follows. In Sections 2, we study
the Hamilton type gradient estimates. In Section 3, we first prove the Li-Yau
type gradient inequality, and then discuss the local Li-Yau type gradient
inequality with lower order term. In Section 4, we give the application to the
Ricci flow.

For readers’ convenience, we will take the same notations as in [13]. We
emphasize that the Laplacian $\Delta$, the gradient $\nabla$, and the norm
$|\cdot|$ appearing above depend on the parameter $t\in[0,T]$. The inner
product $\left<\cdot,\cdot\right>$, the normal vector field
$\frac{\partial}{\partial\nu}$ will be used in the following content also
depend on $t$.

##  2 Gradient estimates of Hamilton type

In this section, we explain how submartingales, related to positive solutions
to the heat equation, can imply the Hamilton type gradient estimates, i.e. the
space-only gradient estimates. For any subset $D\subset M\times[0,T]$ and $f$
defined on $M\times[0,T]$, $\|f\|_{D}:=\sup_{(x,t)\in D}|f|$. The Hamilton
type inequality on compact manifolds is presented as follows.

######  Theorem 2.1 (Gradient inequality of Hamilton type).

Let $M$ be a compact manifold such that ${\mathcal{R}}_{t}\geq-k(t)$ for some
$k\in C([0,T])$. Suppose that $u$ is a positive solution to the heat equation
(1.1) on $M\times[0,T]$. Then for $t\in(0,T]$, we have

$\displaystyle\frac{|\nabla
u|^{2}}{u^{2}}(x,t)\leq\frac{2}{\int_{0}^{t}e^{-\int_{s}^{t}k(r)\mathrm{d}r}\mathrm{d}s}\log\frac{\|u\|_{M\times[0,T]}}{u}.$

The following two lemmas is essential to the proof of Theorem 2.1. First, let
us introduce basic formulas for solutions to the heat equation.

######  Lemma 2.2.

Let $u=u(x,t)$ be a positive solution to (1.1) on $M\times[0,T]$. Then the
following equations hold:

$\displaystyle\left(\frac{1}{2}\Delta-\partial_{t}\right)(u\log
u)=\frac{1}{2}\frac{|\nabla u|^{2}}{u};$
$\displaystyle\left(\frac{1}{2}\Delta-\partial_{t}\right)\frac{|\nabla
u|^{2}}{u}=\frac{1}{u}\left|\nabla^{2}u-\frac{\nabla u\otimes\nabla
u}{u}\right|^{2}+\frac{{\mathcal{R}}_{t}(\nabla u,\nabla u)}{u}.$ (2.1)

The two equalities in Lemma 2.2, which can be checked directly, imply some
inequalities frequently used in the sequel and crucial for our approach. Let
$f=\log u$. We see that

$\frac{1}{u}\left|\nabla^{2}u-\frac{\nabla u\otimes\nabla
u}{u}\right|^{2}=u\left|\nabla^{2}f\right|^{2}\geq\frac{u}{n}(\Delta
f)^{2}=\frac{1}{nu}\left(\Delta u-\frac{|\nabla u|^{2}}{u}\right)^{2}.$ (2.2)

Then, if $\mathcal{R}_{t}\geq-k(t)$ on $M$ for some $k\in C([0,T])$, we have

$\displaystyle\left(\frac{1}{2}\Delta-\partial_{t}\right)\frac{|\nabla
u|^{2}}{u}$ $\displaystyle\geq\frac{1}{nu}\left(\Delta u-\frac{|\nabla
u|^{2}}{u}\right)^{2}+\frac{{\mathcal{R}}_{t}(\nabla u,\nabla u)}{u}$
$\displaystyle\geq-k(t)\frac{|\nabla u|^{2}}{u}.$ (2.3)

Next, we need to construct a (sub)martingale as in [3, Lemma 2.4]. Recall that
$(X_{t}^{T})$ is a $g_{(T-t)}$-Brownian motion on $M$. Let
$\\{P_{s,t}\\}_{0\leq s\leq t\leq T}$ be the associated semigroup. It is easy
to see that $P_{T-t,T}f$ is a solution to the equation (1.1).

######  Lemma 2.3.

Let $u(x,t)=P_{T-t,T}f(x)$ be a positive solution to the heat equation (1.1)
on $M\times[0,T]$. If $\mathcal{R}_{t}\geq-k(t)$ for some $k\in C([0,T])$,
then for any $g_{(T-t)}$-Brownian motion $(X_{t}^{T})$ on $M$, the process

$\displaystyle H_{t}:=h(t)\frac{|\nabla
P_{T-t,T}f|^{2}}{P_{T-t,T}f}(X^{T}_{T-t})+(P_{T-t,T}f\log
P_{T-t,T}f)(X_{T-t}^{T})$ (2.4)

with
$h(t)={\frac{1}{2}\int_{0}^{t}e^{-\int_{s}^{t}k(r)\mathrm{d}r}\mathrm{d}s}$ is
a local supermartingale $($up to lifetime$)$.

###### Proof.

It can be checked directly by the Itô formula, we omit the details. ∎

###### Proof of Theorem 2.1 .

Since $M$ is compact, by Lemma 2.3, the local submartingale $H_{t}$ defined in
(2.4) is a true supermartingale. Then, we have, for $t\in(0,T]$,

$\mathbb{E}(H_{t}|X_{T-t}^{T}=x)\leq\mathbb{E}(H_{0}|X_{T-t}^{T}=x).$

That is

$\left|\frac{\nabla
u}{u}\right|^{2}(x,t)\leq\frac{2}{\int_{0}^{t}e^{-\int_{s}^{t}k(r)\mathrm{d}r}\mathrm{d}s}P_{T-t,T}\left(\frac{f}{P_{T-t,T}f}\log\frac{f}{P_{T-t,T}f}\right)(x).$

By normalizing $f$ as $f^{*}=f/P_{T-t,T}f$, we complete the proof. ∎

Let $\rho_{t}(x,y)$ be the distance between $x\in M$ and $y\in M$ with respect
to the metric $g_{t}$. Fix $x_{0}\in M$ and $\rho>0$. The notation
$B_{\rho,T}$ stands for the set $\\{(x,t)\in M\times[0,T]\ |\
\rho_{t}(x,x_{0})<\rho\\}$. Our next step is to localize the arguments to
cover the solution to the heat equation on $B_{\rho,T}$.

######  Theorem 2.4 (Local gradient inequality of Hamilton type).

Assume that there exist some nonnegative constants $k_{1},k_{2}$ such that

$\displaystyle{\rm Ric}_{t}\geq-k_{1},\ \ \ \partial_{t}g_{t}\geq-k_{2}$ (2.5)

holds on $B_{\rho,T}$. Let $u$ be a solution to the heat equation on
$B_{\rho,T}$, which is positive and continuous on $\overline{B_{\rho,T}}$.
Then, for each $(x,t)\in B_{\rho/2,T}$ and $t\neq 0$, there holds

$\displaystyle\left|\frac{\nabla u}{u}\right|^{2}(x,t)\leq
2\left[\frac{1}{t}+\frac{4\pi^{2}(n+7)}{(4-\pi)^{2}\rho^{2}}+\frac{(\pi^{2}+16)(k_{1}+k_{2})}{(4-\pi)^{2}}\right]\left(4+\log\frac{\|u\|_{\overline{B_{\rho,T}}}}{u}\right)^{2}.$

We want to introduce the Hamilton type inequity on any relatively compact
subset $D$ with nonempty smooth boundary first.

######  Lemma 2.5.

Let $D\subset M\times[0,T]$ be a relatively compact subset with nonempty
smooth boundary. Assume (2.5) holds for some positive constants $k_{1},k_{2}$
on $D$. Let $u$ be a solution to the heat equation on $D$, which is positive
and continuous on $\overline{D}$. Let $\varphi\in C^{1,2}(\bar{D})$ with
$\varphi>0$ and $\varphi|_{\partial D}=0$. Then, for $(x,t)\in D$ and $t\neq
0$, there holds

$\displaystyle\left|\frac{\nabla u}{u}\right|^{2}(x,t)\leq
2\left(\frac{1}{t}+\frac{\sup_{D}\\{7|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}}{\varphi_{t}^{2}(x)}+k_{1}+k_{2}\right)\left(4+\log\frac{\|u\|_{\overline{D}}}{u}\right)^{2}.$
(2.6)

To simplify the proof, we set

$u_{t}=u(X_{T-t}^{T},t),\,\nabla u_{t}=\nabla u(\cdot,t)(X_{T-t}^{T})\
\mbox{and}\ \Delta u_{t}=\Delta u(\cdot,t)(X_{T-t}^{T}).$

Let also

$\displaystyle q(x,t)=\frac{|\nabla u|^{2}}{u}(x,t)\ \ \mbox{and}\ \
q_{t}=q(X_{T-t}^{T},t).$ (2.7)

###### Proof of Lemma 2.5.

The proof is due to that of [3, Theorem 6.1]. We now study the following
process on $D$,

$S_{t}=\frac{|\nabla u_{t}|^{2}}{2u_{t}}-u_{t}(1-\log u_{t})^{2}Z_{t},$

where

$Z_{t}:=\frac{c_{1}}{t}+\frac{c_{2}}{\varphi^{2}_{t}(X^{T}_{T-t})}+c_{3}$

for some constants $c_{1},c_{2},c_{3}>0$, which will be specified later. We
now turn to investigate the martingale property of the process $S_{t}$. It is
easy to see that the process $(X_{T-t}^{T})$ is generated by
$-\frac{1}{2}\Delta$ and it solves the equation

$\mathrm{d}X^{T}_{T-t}=-U^{T}_{T-t}\circ\mathrm{d}B_{t},\
X^{T}_{0}:=\mathbf{p}U^{T}_{0}=x\in D,$

up to the life time, where $U^{T}_{T-t}$ is the horizontal process of
$X^{T}_{T-t}$. Assume that $0<u\leq e^{-3}$ (the assumption will be removed in
Lemma 2.5 through replacing $u$ by $e^{-3}u/\|u\|_{D}$). Using the fact that

$\nabla(u_{t}(1-\log u_{t})^{2})=(\log^{2}u_{t}-1)\nabla u_{t}\ \ \mbox{and}\
\ \mathrm{d}(u_{t}(1-\log u_{t})^{2}){=}-q_{t}\log u_{t}\mathrm{d}t,$

we get (the indicator “$\ \stackrel{{\scriptstyle m}}{{}}\ $” stands for the
modula differential of local martingales)

$\displaystyle\mathrm{d}S_{t}=$
$\displaystyle\frac{1}{2}\mathrm{d}\left\\{\frac{|\nabla
u_{t}|^{2}}{u_{t}}\right\\}-u_{t}(1-\log
u_{t})^{2}\mathrm{d}Z_{t}-Z_{t}\mathrm{d}(u_{t}(1-\log u_{t})^{2})$
$\displaystyle+2c_{2}(1-\log u_{t})(1+\log
u_{t})\varphi_{t}^{-3}(X_{T-t}^{T})\left<\nabla
u_{t},\nabla\varphi_{t}(X^{T}_{T-t})\right>\mathrm{d}t$
$\displaystyle\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\frac{1}{2}(k_{1}+k_{2})q_{t}\mathrm{d}t+u_{t}(1-\log
u_{t})^{2}\left[c_{1}t^{-2}+c_{2}c_{\varphi}(X_{T-t}^{T},t)\varphi_{t}^{-4}(X_{T-t}^{T})\right]\mathrm{d}t+\log
u_{t}q_{t}Z_{t}\mathrm{d}t$ $\displaystyle+2c_{2}(1-\log u_{t})(1+\log
u_{t})\varphi_{t}^{-3}(X^{T}_{T-t})\left<\nabla
u_{t},\nabla\varphi_{t}(X_{T-t}^{T})\right>\mathrm{d}t,$

where
$c_{\varphi}(x,t)=\left[3|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\right](x).$
More, we have $2\log u_{t}\leq-3(1-\log u_{t})/2$ from $u\leq e^{-3}$, which,
together with $|1+\log u_{t}|\leq 1-\log u_{t}$, yields

$\displaystyle\mathrm{d}S_{t}\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle(\log
u_{t}-1)\bigg{\\{}\frac{1}{2}\left(\frac{3}{2}Z_{t}-k_{1}-k_{2}\right)q_{t}-u_{t}(1-\log
u_{t})\left[c_{1}t^{-2}-c_{2}c_{\varphi}(X_{T-t}^{T},t)\varphi_{t}^{-4}(X_{T-t}^{T})\right]$
$\displaystyle-c_{2}2\sqrt{2}(1-\log
u_{t})\varphi_{t}^{-2}(X_{T-t}^{T})|\nabla\varphi_{t}(X_{T-t}^{T})|\sqrt{u_{t}}\times\varphi_{t}^{-1}(X_{T-t}^{T})\sqrt{\frac{q_{t}}{2}}\bigg{\\}}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle(\log
u_{t}-1)\bigg{\\{}\frac{1}{2}\left(\frac{3}{2}Z_{t}-k_{1}-k_{2}\right)q_{t}-u_{t}(1-\log
u_{t})\left[c_{1}t^{-2}-c_{2}c_{\varphi}(X_{T-t}^{T},t)\varphi_{t}^{-4}(X_{T-t}^{T})\right]$
$\displaystyle-4c_{2}(1-\log
u_{t})^{2}\varphi_{t}^{-4}(X_{T-t}^{T})|\nabla\varphi_{t}(X_{T-t}^{T})|^{2}u_{t}-c_{2}\varphi_{t}^{-2}(X_{T-t}^{T})\frac{q_{t}}{4}\bigg{\\}}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle(\log u_{t}-1)\bigg{\\{}-u_{t}(1-\log
u_{t})^{2}\times\left[c_{1}t^{-2}+c_{2}\left(c_{\varphi}(X_{T-t}^{T},t)+4|\nabla\varphi_{t}(X_{T-t}^{T})|^{2}\right)\varphi_{t}^{-4}(X_{T-t}^{T})\right]$
$\displaystyle\qquad\qquad\quad+(Z_{t}-k_{1}-k_{2})\frac{1}{2}q_{t}\bigg{\\}}\mathrm{d}t.$

Let $c_{1}=1,c_{2}=\sup_{D}\\{c_{\varphi}+4|\nabla\varphi_{t}|^{2}\\}$ and
$c_{3}=k_{1}+k_{2}$. We conclude

$\displaystyle\mathrm{d}S_{t}$ $\displaystyle\stackrel{{\scriptstyle
m}}{{\leq}}-(1-\log
u_{t})(Z_{t}-k_{1}-k_{2})\left[\frac{1}{2}q_{t}-u_{t}(1-\log
u_{t})^{2}Z_{t}\right]\mathrm{d}t$ $\displaystyle=-(1-\log
u_{t})(Z_{t}-k_{1}-k_{2})S_{t}\mathrm{d}t.$

Now we consider the process $\\{X^{T}_{T-s}\\}_{s\in[0,t]}$ starting from $x$
at time $t$. This process has nonpositive drift on $\\{S_{s}\geq 0\\}$. On the
other hand, $S_{s}$ converges to $-\infty$ as $s\rightarrow 0\vee\tau(x)$,
where $\tau(x):=\sup\\{s\in[0,t]:X_{T-s}^{T}\notin D,X_{T-t}^{T}=x\\},\
\sup\varnothing=0$. Therefore, $S_{t}\leq 0$, i.e.

$\frac{|\nabla u_{t}|^{2}}{2u_{t}}\leq(1-\log u_{t})^{2}Z_{t}.$

Replacing $u$ by $e^{-3}u/\|u\|_{D}$, we complete the proof. ∎

###### Proof of Theorem 2.4.

To specify the constants of (2.6) in terms of $\varphi$, an explicit choice
for $\varphi$ has to be done. For any $y\in B_{\rho,T}$, i.e.
$\rho_{t}(x_{0},y)\leq\rho$, $t\in[0,T]$,

$\displaystyle\varphi(t,y):=\cos\frac{\pi\rho_{t}(x_{0},y)}{2\rho}.$ (2.8)

Then, $\varphi$ is nonnegative and bounded by $1$, and $\varphi$ vanishes on
$\partial B_{\rho,T}$. Fix $(x,t)\in B_{\rho/2,T}$, we have

$\varphi_{t}(x)=\varphi(t,x)=\cos\frac{\pi\rho_{t}(x_{0},x)}{2\rho}\geq
1-\frac{\pi\rho_{t}(x_{0},x)}{2\rho}\geq 1-\frac{\pi}{4}.$ (2.9)

By convention, $\Delta\varphi_{t}^{-2}=0$, when $\varphi_{t}^{-2}$ is not
differentiable. As a consequence, all the estimates in Lemma 2.5 still hold
true with $\varphi$ defined by (2.8).

Now, we derive explicit estimates for the constants. To this end, we observe
that

$\|\nabla\varphi_{t}\|_{B_{\rho,T}}\leq\frac{\pi}{2\rho}.$ (2.10)

Since ${\rm Ric}_{t}\geq-k_{1}$, $\partial_{t}g_{t}\geq-k_{2}$ for
$k_{1},k_{2}>0$, by the index lemma, we have

$\displaystyle-(\Delta-2\partial_{t})\varphi_{t}=$
$\displaystyle\sin\frac{\pi\rho_{t}(x_{0},y)}{2\rho}\frac{\pi}{2\rho}(\Delta-2\partial_{t})\rho_{t}(x_{0},\cdot)(y)+\cos\frac{\pi\rho_{t}(x_{0},y)}{2\rho}\cdot\frac{\pi^{2}}{4\rho^{2}}|\nabla\rho_{t}|^{2}$
$\displaystyle\leq$
$\displaystyle\sin\frac{\pi\rho_{t}(x_{0},y)}{2\rho}\frac{\pi}{2\rho}\left[\frac{n-1}{\rho_{t}(x_{0},y)}+(k_{1}+k_{2})\rho_{t}(x_{0},y)\right]+\frac{\pi^{2}}{4\rho^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{\pi^{2}n}{4\rho^{2}}+\frac{\pi}{2}(k_{1}+k_{2}).$

Thus, we further have

$\sup_{B_{\rho,T}}\\{7|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}\leq\frac{\pi^{2}(n+7)}{4\rho^{2}}+\frac{\pi}{2}(k_{1}+k_{2}).$

Combining this with (2.6) and (2.9), we complete the proof. ∎

##  3  Li-Yau type gradient inequalities

This section is devoted to extending the argument of the Li-Yau type
inequalities on Riemannian manifolds to our setting.

###  3.1 Li-Yau type inequalities

First, we present the local version of the Li-Yau gradient inequality as
follows.

######  Theorem 3.1 (Local gradient inequality of Li-Yau type).

Assume that there exist some nonnegative constants $k_{1},k_{2},k_{3},k_{4}$
such that

$\displaystyle{\rm Ric}_{t}\geq-k_{1},\ \ \ -k_{2}\leq\partial_{t}g_{t}\leq
k_{3},\ \ \ |\nabla(\partial_{t}g_{t})|\leq k_{4}$ (3.1)

holds on $B_{\rho,T}$. Let $u$ be a solution to the heat equation on
$B_{\rho,T}$, which is positive and continuous on $\overline{B_{\rho,T}}$. For
any $\alpha>1$ and $t>0$,

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{\Delta u}{u}\leq
n\alpha^{2}\bigg{[}$
$\displaystyle\frac{2}{t}+\frac{8\pi^{2}}{(4-\pi)^{2}\rho^{2}}\left(n+3+\frac{\alpha^{2}n}{\alpha-1}\right)+\frac{16{\pi}(k_{1}+k_{2})}{(4-\pi)^{2}}+\max\\{k_{2},k_{3}\\}+k_{3}$
$\displaystyle+\sqrt{{2k_{4}}}+\frac{k_{1}+k_{4}}{\alpha-1}\bigg{]}$

holds on $B_{\rho/2,T}$.

Parallel to Theorem 2.4, we need to consider the local Li-Yau inequality on
some subset $D$.

######  Lemma 3.2.

Let $D\subset M\times[0,T]$ be a relatively compact subset with nonempty
smooth boundary. Assume (3.1) holds for some non-negative constants
$k_{1},k_{2},k_{3},k_{4}$ on $D$. Let $u$ be a solution to the heat equation
on $D$, which is positive and continuous on $\overline{D}$. Let $\varphi\in
C^{1,2}(\overline{D})$ with $\varphi>0$ and $\varphi|_{\partial D}=0$. For any
$\alpha>1$ and $t>0$, we have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{\Delta u}{u}\leq
n\alpha^{2}\bigg{(}$
$\displaystyle\frac{2}{t}+\frac{2C_{\varphi,\alpha,n}}{\varphi_{t}^{2}(x)}+\max\\{k_{2},k_{3}\\}+k_{3}+\sqrt{{2k_{4}}}+\frac{k_{1}+k_{4}}{\alpha-1}\bigg{)}$

holds on $D$, where
$C_{\varphi,\alpha,n}:=\sup_{D}\\{(3+\alpha^{2}(\alpha-1)^{-1}n)|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}$.

###### Proof.

Define

$\displaystyle
Y_{t}=c_{1}t^{-1}+c_{2}\varphi_{t}^{-2}(X_{T-t}^{T})+c_{3}\max\\{k_{2},k_{3}\\}+c_{4}\sqrt{k_{4}}+\frac{k_{1}+(\alpha-1)k_{3}+k_{4}}{\alpha-1},\
\alpha>1,$

where $c_{1},c_{2},c_{3},c_{4}>0$ are constants, which will be specified
later. Let $h_{t}$ be the solution of $\dot{h}_{t}=h_{t}Y_{t},\ h_{T}=1$.
Consider the following process,

$\displaystyle S_{t,\alpha}:=h_{t}(q_{t}-\alpha\Delta u_{t})-\beta
nu_{t}\dot{h}_{t}\equiv h_{t}\left(q_{t}-\alpha\Delta u_{t}-n\beta
u_{t}Y_{t}\right),$

where $\beta>0$ will also be specified later and $q_{t}$ is defined in (2.7).
We first investigate the martingale property of $S_{t,\alpha}$. By (2.1) and
(2.2), we find

$\mathrm{d}(h_{t}q_{t})\stackrel{{\scriptstyle
m}}{{\leq}}\bigg{[}-h_{t}u_{t}|\nabla^{2}f|^{2}-\dot{h}_{t}q_{t}+h_{t}u_{t}{\rm
Ric}_{t}(\nabla f,\nabla f)+h_{t}u_{t}\partial_{t}g_{t}(\nabla f,\nabla
f)\bigg{]}\mathrm{d}t,$

By this and observing from [13, Lemma 3.2],

$\displaystyle\mathrm{d}(\Delta u_{t})$ $\displaystyle\stackrel{{\scriptstyle
m}}{{=}}\left[-u_{t}\left<\partial_{t}g_{t},\nabla^{2}f\right>-u_{t}\partial_{t}g_{t}(\nabla
f,\nabla f)-u_{t}\left<{\rm div}(\partial_{t}g_{t})-\frac{1}{2}\nabla({\rm
tr}_{g}(\partial_{t}g_{t})),\nabla f\right>\right](X_{T-t}^{T},t)\mathrm{d}t$
$\displaystyle\geq\left[-u_{t}\left<\partial_{t}g_{t},\nabla^{2}f\right>-u_{t}\partial_{t}g_{t}(\nabla
f,\nabla f)-2\sqrt{n}k_{4}u_{t}|\nabla f|\right](X_{T-t}^{T},t)\mathrm{d}t$
$\displaystyle\geq\left[-u_{t}\left<\partial_{t}g_{t},\nabla^{2}f\right>-u_{t}\partial_{t}g_{t}(\nabla
f,\nabla f)-\alpha nk_{4}u_{t}|\nabla
f|^{2}-\alpha^{-1}k_{4}u_{t}\right](X_{T-t}^{T},t)\mathrm{d}t,$

we have

$\displaystyle\mathrm{d}S_{t,\alpha}{=}$
$\displaystyle\mathrm{d}h_{t}q_{t}-n\beta
u_{t}\mathrm{d}(h_{t}Y_{t})-n\beta\mathrm{d}[u,\dot{h}]_{t}-\alpha\mathrm{d}h_{t}\Delta
u_{t}$ $\displaystyle\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\big{[}-h_{t}u_{t}|\nabla^{2}f|^{2}+\alpha
h_{t}u_{t}\left<\partial_{t}g_{t},\nabla^{2}f\right>-\alpha\dot{h}_{t}(\Delta
u_{t})\big{]}\mathrm{d}t$
$\displaystyle+(\dot{h}_{t}+k_{1}h_{t}+k_{4}h_{t}+(\alpha-1)k_{3}h_{t})q_{t}\mathrm{d}t-n\beta
u_{t}h_{t}Y_{t}^{2}\mathrm{d}t$ $\displaystyle+n\beta
u_{t}h_{t}\big{[}c_{1}t^{-2}+c_{2}c_{\varphi}(X_{T-t}^{T},t)\varphi^{-4}_{t}(X_{T-t}^{T})+\beta^{-1}k_{4}\alpha^{2}\big{]}\mathrm{d}t-nc_{2}\beta
h_{t}\mathrm{d}[u,\varphi_{\cdot}^{-2}(X^{T}_{T-\cdot})]_{t},$

where
$c_{\varphi}(x,t)=\left[3|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\right](x).$
As $\Delta f_{t}=\frac{1}{u_{t}}(\Delta u_{t}-q_{t})$, we have

$\displaystyle\mathrm{d}S_{t,\alpha}\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\big{[}-h_{t}u_{t}|\nabla^{2}f|^{2}(X_{T-t}^{T},t)+\alpha
h_{t}u_{t}\left<\partial_{t}g_{t},\nabla^{2}f\right>(X_{T-t}^{T},t)-\alpha
h_{t}Y_{t}u_{t}(\Delta f_{t})\big{]}\mathrm{d}t$
$\displaystyle+(k_{1}+(\alpha-1)k_{3}+k_{4})h_{t}q_{t}\mathrm{d}t-(\alpha-1)\dot{h}_{t}q_{t}\mathrm{d}t-n\beta
u_{t}h_{t}Y_{t}^{2}\mathrm{d}t$ $\displaystyle+n\beta
u_{t}h_{t}\big{[}c_{1}t^{-2}+c_{2}c_{\varphi}(X_{T-t}^{T},t)\varphi^{-4}_{t}(X_{T-t}^{T})+\beta^{-1}k_{4}\alpha^{2}\big{]}\mathrm{d}t$
$\displaystyle-nc_{2}\beta
h_{t}\mathrm{d}[u,\varphi_{\cdot}^{-2}(X^{T}_{T-\cdot})]_{t}.$ (3.2)

For any $a,b>0$ such that $a+b=\alpha^{-1}$,

$\displaystyle|\nabla^{2}f|^{2}-\alpha\left<\partial_{t}g_{t},\nabla^{2}f\right>=$
$\displaystyle\left\\{\left(a\alpha+b\alpha\right)|\nabla^{2}f|^{2}-\alpha\right<\partial_{t}g_{t},\nabla^{2}f\left>\right\\}$
$\displaystyle=$
$\displaystyle\bigg{[}a\alpha|\nabla^{2}f|^{2}+\alpha\left|\sqrt{b}\nabla^{2}f-\frac{1}{2\sqrt{b}}\partial_{t}g_{t}\right|^{2}-\frac{\alpha}{4b}|\partial_{t}g_{t}|^{2}\bigg{]}$
$\displaystyle\geq$
$\displaystyle\bigg{[}a\alpha|\nabla^{2}f|^{2}-\frac{\alpha}{4b}|\partial_{t}g_{t}|^{2}\bigg{]}\geq\frac{a\alpha}{n}(\Delta
f)^{2}-\frac{\alpha n}{4b}\max\\{k_{2}^{2},k_{3}^{2}\\}$ (3.3)

at $(x,t)\in D$. So,

$\displaystyle|\nabla^{2}f|^{2}-\alpha\left<\partial_{t}g_{t},\nabla^{2}f\right>-\alpha
Y_{t}(\Delta f)$ $\displaystyle\geq\frac{a\alpha}{n}(\Delta
f)^{2}-\frac{\alpha
b}{4b}\max\left\\{k_{2}^{2},k_{3}^{2}\right\\}-\alpha(\Delta f)Y_{t}$
$\displaystyle\geq-\frac{n\alpha}{4a}Y_{t}^{2}-\frac{\alpha
n}{4b}\max\\{k_{2}^{2},k_{3}^{2}\\}.$ (3.4)

In addition, for the last term in (3.1), we have

$\displaystyle-n\beta$ $\displaystyle
c_{2}h_{t}\mathrm{d}[u,\varphi_{\cdot}^{-2}(X_{T-\cdot}^{T})]_{t}=2n\beta
c_{2}h_{t}\varphi_{t}^{-3}\left<\nabla
u_{t},\nabla\varphi_{t}(X^{T}_{T-t})\right>\mathrm{d}t$ $\displaystyle\leq
2nc_{2}h_{t}\left[\varphi^{-1}_{t}((\alpha-1)^{-1}nu_{t})^{-1/2}|\nabla
u_{t}|\times\beta((\alpha-1)^{-1}nu_{t})^{1/2}\varphi_{t}^{-2}|\nabla\varphi_{t}|\right](X^{T}_{T-t})\mathrm{d}t$
$\displaystyle\leq\left[(\alpha-1)c_{2}h_{t}\varphi_{t}^{-2}u_{t}^{-1}|\nabla
u_{t}|^{2}+n^{2}c_{2}h_{t}\beta^{2}(\alpha-1)^{-1}u_{t}\varphi_{t}^{-4}|\nabla\varphi_{t}|^{2}\right](X^{T}_{T-t})\mathrm{d}t$
$\displaystyle\leq\left\\{[(\alpha-1)\dot{h}_{t}-(k_{1}+k_{4}+(\alpha-1)k_{3})h_{t}]q_{t}+c_{2}(\alpha-1)^{-1}\beta^{2}n^{2}u_{t}h_{t}\varphi_{t}^{-4}|\nabla\varphi_{t}|^{2}(X^{T}_{T-t})\right\\}\mathrm{d}t.$
(3.5)

Combining this with (3.1), (3.1) and (3.1), we arrive at

$\displaystyle\mathrm{d}S_{t,\alpha}\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\left[-nh_{t}Y_{t}^{2}u_{t}\left(\beta-\frac{\alpha}{4a}\right)+\frac{n\alpha}{4b}u_{t}h_{t}\max\\{k_{2}^{2},k_{3}^{2}\\}\right]\mathrm{d}t$
$\displaystyle+nu_{t}h_{t}\left(\beta c_{1}t^{-2}+\beta
c_{2}C_{\varphi,\alpha,\beta,n}\varphi_{t}^{-4}(X_{T-t}^{T})+\alpha^{2}k_{4}\right)\mathrm{d}t,$

where $a+b=\alpha^{-1}$ and
$C_{\varphi,\alpha,\beta,n}=\sup_{D}\\{(3+\beta(\alpha-1)^{-1}n)|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}$.
Let

$\displaystyle\left(\beta-\frac{\alpha}{4a}\right)c_{1}^{2}\geq\beta c_{1},\ \
\ \ \left(\beta-\frac{\alpha}{4a}\right)c_{2}^{2}-\beta
c_{2}C_{\varphi,\alpha,\beta,n}\geq 0,\ \ $
$\displaystyle\left(\beta-\frac{\alpha}{4a}\right)c_{3}^{2}-\frac{\alpha}{4b}\geq
0,\ \ \left(\beta-\frac{\alpha}{4a}\right)c_{4}^{2}-\alpha^{2}\geq 0.$ (3.6)

Using

$Y_{t}^{2}\geq
c_{1}^{2}t^{-2}+c_{2}^{2}\varphi_{t}^{-4}(X_{T-t}^{T})+c_{3}^{2}\max\\{k_{2}^{2},k_{3}^{2}\\}+c_{4}^{2}k_{4},$

we obtain that the right side of (LABEL:estimate-S1-3) is non-positive.
Consider the process $\\{X_{T-s}^{T}\\}_{s\in[0,t]}$ with $X_{T-t}^{T}=x$ and
$(x,t)\in D$. Let $\tau(x):=\sup\\{s<t:X_{T-s}^{T}\notin D,s\geq 0\\}$ and
$\sup\varnothing=0$. Since $q_{t}-\alpha\Delta u_{t}-nu_{t}Y_{t}$ converges to
$-\infty$ as $t$ tends to $0\vee\tau(x)$, the process $S_{t,\alpha}$ has non-
positive drift, i.e. $S_{t,\alpha}(x)\leq 0$, which implies

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{\Delta u}{u}\leq
n\beta\left(\frac{c_{1}}{t}+\frac{c_{2}}{\varphi_{t}^{2}(x)}+c_{3}\max\\{k_{2},k_{3}\\}+c_{4}\sqrt{k_{4}}+\frac{k_{1}+k_{4}}{\alpha-1}+k_{3}\right).$
(3.7)

Set $a=b=\frac{1}{2\alpha}$ and $\beta=\frac{\alpha}{2a}=\alpha^{2}>0$. Let

$C_{\varphi,\alpha,n}:=\sup_{D}\\{(3+\alpha^{2}(\alpha-1)^{-1}n)|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta+2\partial_{t})\varphi_{t}\\}.$

Then, from (3.1), we select $c_{1},c_{2},c_{3},c_{4}$ such that

$c_{3}=1,c_{2}=2C_{\varphi,\alpha,n},c_{1}=2,\ c_{4}=\sqrt{{2}},$

we complete the proof by taking these constants $c_{1},c_{2},c_{3},c_{4}$ into
(3.7). ∎

###### Proof of Theorem 3.1.

When $D=B_{\rho,T}$, the function $\varphi$ is chosen as in (2.8). Moreover,
let ${\rm Cut}_{t}(x_{0})$ be the set of the $g_{t}$ cut-locus of $x_{0}$ on
$M$. Since the time spent by $X_{T-t}^{T}$ on $\bigcup_{t\in[0,T]}{\rm
Cut}_{t}(x_{0})$ is a.s. zero (see [9]), the differential of the brackets
$[\varphi_{\cdot}(X_{T-\cdot}^{T}),u_{\cdot}]_{t}$ may be taken as 0 at points
where $\varphi_{t}^{-2}$ is not differentiable. Therefore,

$\displaystyle C_{\varphi,\alpha,n}$
$\displaystyle=\sup_{B_{\rho,T}}\\{(3+\alpha^{2}(\alpha-1)^{-1}n)|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}$
$\displaystyle\leq\frac{\pi^{2}(n+3+\alpha^{2}(\alpha-1)^{-1}n)}{4\rho^{2}}+\frac{\pi}{2}(k_{1}+k_{2}),$

combining this with Lemma 3.2, we complete the proof. ∎

######  Theorem 3.3 (Gradient inequality of Li-Yau type).

Let $M$ be a compact manifold. Assume (3.1) holds for some nonnegative
constants $k_{1},k_{2},k_{3},k_{4}$ on $M\times[0,T]$. Let $u$ be the positive
solution of the heat equation (1.1) on $M\times[0,T]$. For any $\alpha>1$, we
have

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{\Delta u}{u}\leq
n\alpha^{2}\bigg{(}$
$\displaystyle\frac{2}{t}+\max\\{k_{2},k_{3}\\}+k_{3}+\sqrt{2k_{4}}+\frac{k_{1}+k_{4}}{\alpha-1}\bigg{)}$
(3.8)

holds on $M\times[0,T]$.

###### Proof.

Define

$\tilde{Y}_{t}=c_{1}t^{-1}+c_{2}\max\\{k_{2},k_{3}\\}+c_{3}\sqrt{k_{4}}+k_{3}+\frac{k_{1}+k_{4}}{\alpha-1},$

where $c_{1},c_{2},c_{3}$ are constants, which will be specified later. Let
$h_{t}$ be the solution of

$\dot{h}_{t}=h_{t}Y_{t},\ \ h_{T}=1.$

Consider the following process

$\tilde{S}_{t,\alpha}:=h_{t}(q_{t}-\alpha\Delta u_{t}-n\beta
u_{t}\tilde{Y}_{t}),$

where $\beta>0$ will be also specified later. By a similar discussion as in
the proof of Lemma 3.2, we obtain

$\displaystyle\mathrm{d}\tilde{S}_{t,\alpha}\stackrel{{\scriptstyle
m}}{{\leq}}$
$\displaystyle\left[-nh_{t}Y_{t}^{2}u_{t}\left(\beta-\frac{\alpha}{4a}\right)+\frac{n\alpha}{4b}u_{t}h_{t}\max\\{k_{2}^{2},k_{3}^{2}\\}\right]\mathrm{d}t$
$\displaystyle+nu_{t}h_{t}\left(\beta
c_{1}t^{-2}+\alpha^{2}k_{4}\right)\mathrm{d}t+(k_{1}+(\alpha-1)k_{3}+k_{4})h_{t}q_{t}\mathrm{d}t-(\alpha-1)\dot{h}_{t}q_{t}\mathrm{d}t$
$\displaystyle\leq$
$\displaystyle\left[-nh_{t}Y_{t}^{2}u_{t}\left(\beta-\frac{\alpha}{4a}\right)+\frac{n\alpha}{4b}u_{t}h_{t}\max\\{k_{2}^{2},k_{3}^{2}\\}\right]\mathrm{d}t+nu_{t}h_{t}\left(\beta
c_{1}t^{-2}+\alpha^{2}k_{4}\right)\mathrm{d}t.$

The rest part is similar to that of Lemma 3.2, we omit it here. ∎

###  3.2 Li-Yau type inequality with lower order term

Now, we turn to studying the Li-Yau type inequality with the lower order term.
The main result of this subsection is the following.

######  Theorem 3.4.

Under the same condition as in Theorem 2.4. Then, for $t>0$,

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\frac{\Delta u}{u}\leq$
$\displaystyle\frac{2n}{t}+\frac{8n\pi^{2}(n+3)}{(4-\pi)^{2}\rho^{2}}+\frac{16n\pi(k_{1}+k_{2})}{(4-\pi)^{2}}+\max\\{k_{2},k_{3}\\}n+\sqrt{2k_{4}}n$
$\displaystyle+\bigg{[}\frac{8\pi
n}{(4-\pi)^{2}\rho}+\sqrt{2n(k_{1}+k_{4})}\bigg{]}\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{B_{\rho,T}}$

holds on $B_{\rho/2,T}$.

To prove this theorem, we first give a more general result.

######  Lemma 3.5 (Local Li-Yau type with lower order term).

Under the same condition as in Lemma 2.5. Then, for $t>0$,

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\frac{\Delta u}{u}\leq$
$\displaystyle\frac{2n}{t}+\frac{2n\sup_{D}\\{3|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}}{\varphi_{t}^{2}(x)}+\max\\{k_{2},k_{3}\\}n+\sqrt{2k_{4}}n$
$\displaystyle+\bigg{(}4n\frac{\|\varphi_{\cdot}\nabla^{\cdot}\varphi_{\cdot}\|_{D}}{\varphi_{t}^{2}(x)}+\sqrt{2n(k_{1}+k_{4})}\bigg{)}\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{D}$

holds on $D$.

###### Proof.

Let

$\tilde{S}_{t}=\frac{|\nabla u_{t}|^{2}}{u_{t}}-\Delta
u_{t}-nu_{t}\tilde{Z}_{t},$

where $\tilde{Z}_{t}=c_{1}t^{-1}+c_{2}\varphi_{t}^{-2}(X_{T-t}^{T})+c_{3}$
with constants $c_{1},c_{2},c_{3}>0$ to be specified later. Let

$c_{\varphi}:=\sup_{D}\\{3|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}.$

By a similar calculation as in the proof of Lemma 3.2, we have

$\displaystyle\mathrm{d}\tilde{S}_{t}\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\left[-\frac{a}{nu_{t}}\left(\frac{|\nabla
u_{t}|^{2}}{u_{t}}-\Delta u_{t}\right)^{2}+(k_{1}+k_{4})\frac{|\nabla
u_{t}|^{2}}{u_{t}}+\frac{n}{4b}u_{t}\max\\{k_{2}^{2},k_{3}^{2}\\}+nu_{t}k_{4}\right]\mathrm{d}t$
$\displaystyle+nu_{t}(c_{1}t^{-2}+c_{2}c_{\varphi}\varphi_{t}^{-4}(X_{T-t}^{T}))\mathrm{d}t-nc_{2}\mathrm{d}[\varphi_{\cdot}^{-2}(X_{T-{\cdot}}^{T}),u]_{t},$

where $a+b=1$. Since

$\displaystyle-nc_{2}\mathrm{d}[\varphi_{t}^{-2}(X_{T-t}^{T}),u]_{t}=$
$\displaystyle 2nc_{2}\varphi_{t}^{-4}(X_{T-t}^{T})\left<\frac{\nabla
u_{t}}{u_{t}},\varphi_{t}\nabla\varphi_{t}(X_{T-t}^{T})\right>u_{t}\mathrm{d}t$
$\displaystyle\leq$ $\displaystyle
2nc_{2}u_{t}\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{D}\|\varphi_{\cdot}\nabla^{\cdot}\varphi_{\cdot}\|_{D}\varphi_{t}^{-4}(X_{T-t}^{T})\mathrm{d}t,$

we arrive at

$\displaystyle\mathrm{d}\tilde{S}_{t}\stackrel{{\scriptstyle m}}{{\leq}}$
$\displaystyle\bigg{\\{}-\frac{a}{nu_{t}}\left(\frac{|\nabla
u_{t}|^{2}}{u_{t}}-\Delta
u_{t}\right)^{2}+nu_{t}\bigg{[}\frac{c_{1}}{t^{2}}+\frac{c_{2}}{\varphi_{t}^{4}(X_{T-t}^{T})}\times\bigg{(}c_{\varphi}+2\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{D}\|\varphi_{\cdot}\nabla^{\cdot}\varphi_{\cdot}\|_{D}\bigg{)}\bigg{]}\bigg{\\}}\mathrm{d}t$
$\displaystyle+nu_{t}\frac{k_{1}+k_{4}}{n}\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|^{2}_{D}\mathrm{d}t+\frac{1}{4b}nu_{t}\max\\{k_{2}^{2},k_{3}^{2}\\}\mathrm{d}t+nu_{t}k_{4}\mathrm{d}t.$

Then, letting $c_{1}=\frac{1}{a}$,
$c_{2}=\frac{1}{a}\left[c_{\varphi}+2\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{D}\|\varphi_{\cdot}\nabla^{(T-{\cdot})}\varphi_{\cdot}\|_{D}\right]$,
and

$c_{3}=\sqrt{\frac{k_{1}+k_{4}}{an}}\left\|\frac{|\nabla^{\cdot}u|}{u}\right\|_{D}+\sqrt{\frac{1}{4ab}}\max\\{k_{2},k_{3}\\}+\sqrt{\frac{k_{4}}{a}},$

we obtain

$\mathrm{d}\tilde{S}_{t}\leq-\frac{a\tilde{S}_{t}}{nu_{t}}\left(\frac{|\nabla
u_{t}|^{2}}{u_{t}}-\Delta u_{t}+nu_{t}Z_{t}\right)\mathrm{d}t.$

Then, on $\\{\tilde{S}_{t}\geq 0\\}$, the process $\tilde{S}_{t}$ has
nonpositive drift. Consider $\\{X_{T-s}^{T}\\}_{s\in[0,t]}$ starting from $x$
at $s=t$. Since $\tilde{S}_{t}$ goes to $-\infty$ as $s\rightarrow
0\wedge\tau(x)$. We have $\tilde{S}_{t}\leq 0$. Now, setting
$a=b=\frac{1}{2}$, we complete the proof. ∎

###### Proof of Theorem 3.4.

When $D=B_{\rho,T}$, the function $\varphi$ is chosen in (2.8). Since

$\displaystyle c_{\varphi}$
$\displaystyle=\sup_{B_{\rho,T}}\\{3|\nabla\varphi_{t}|^{2}-\varphi_{t}(\Delta-2\partial_{t})\varphi_{t}\\}\leq\frac{\pi^{2}(n+3)}{4\rho^{2}}+\frac{\pi}{2}(k_{1}+k_{2}),$
(3.9)

combining this with (2.10) and (3.8), we complete the proof. ∎

##  4 Applications to Ricci flow

In this section, we apply our results to the special case of the Ricci flow
(1.2). In this case, when $M$ is compact, Theorem 2.1 reduces to [6, 14]. That
is

$\frac{|\nabla
u|}{u}\leq\sqrt{\frac{2}{t}\log\frac{\|u\|_{M\times[0,T]}}{u}}.$

Remark. When the manifold $M$ carries boundary, we turn to consider the system
as follows: for $\lambda\geq 0$,

$\begin{cases}\frac{\partial}{\partial t}g(x,t)=-{\rm Ric}(x,t),\ &\ \
(x,t)\in M\times[0,T];\\\ \mathbb{I}_{t}=\lambda,\ &\ \ x\in\partial
M,\end{cases}$ (4.1)

where $\mathbb{I}_{t}$ is the second fundamental form w.r.t. the metric
$g_{t}$. Shen [12] give the proof of the short time exitance of the solution.
Suppose a smooth positive function $u:M\times[0,T]\rightarrow\mathbb{R}$
satisfies the heat equation

$\begin{cases}\frac{\partial}{\partial t}u(x,t)=\frac{1}{2}\Delta u(x,t),&\ \
(x,t)\in M\times[0,T];\\\ \frac{\partial}{\partial\nu}u(x,t)=0,\ &\ \
x\in\partial M,\end{cases}$ (4.2)

where $\frac{\partial}{\partial\nu}$ is the inward unit normal vector field of
the boundary associated with $g_{t}$. Let $X_{t}^{T}$ be a
$g_{(T-t)}$-Brownian motion and $P_{s,t}$ be the associated semigroup (see
[7]). Since $\mathbb{I}_{t}=\lambda\geq 0$ and
$\frac{\partial}{\partial\nu}P_{T-t,T}f=0$ (see [7, Theorem 2.1(2)]), the
process $H_{t}$, constructed as in Lemma 2.3 by using the reflecting
$g_{(T-t)}$-Brownian motion, is also a submartingale for the system
(4.1)-(4.2). Then, the conclusion in Theorem 2.1 still holds true.

Next, we consider the Li-Yau type gradient inequality on a compact manifold
carrying the Ricci flow.

######  Theorem 4.1.

Suppose the manifold $M$ is compact and $0\leq{\rm Ric}_{t}\leq k$,
$t\in(0,T]$. Let $(M,g_{t})_{t\in[0,T]}$ be a solution of the Ricci flow.
Assume a smooth positive function $u:M\times[0,T]\rightarrow\mathbb{R}$
satisfies the heat equation (1.1). Then the following estimates

$\displaystyle\frac{|\nabla u|^{2}}{u^{2}}-\frac{\Delta u}{u}\leq
kn+\frac{2n}{t}$ (4.3)

hold for all $M\times(0,T]$.

###### Proof.

Note that when $\\{g_{t}\\}_{t\in[0,T]}$ is a Ricci flow, the second Bianchi
identity says that

${\rm div}(\partial_{t}g_{t})-\frac{1}{2}\nabla({\rm
tr}_{g}(\partial_{t}g_{t}))=0.$

So we do not need the condition $|\nabla h|\leq k_{4}$ to deal with
$\mathrm{d}(\Delta u_{t})$. Consider the process $\hat{S}_{t}$ constructed in
(1.3). By a similar discussion as in the proof of Theorem 3.3, we obtain (4.3)
directly. Here, we omit the details. ∎

Remark. (1) In [4], the authors gave the Li-Yau type gradient estimate on the
compact manifold as follows,

$\frac{|\nabla u|^{2}}{u^{2}}-\frac{\Delta u}{u}\leq 2kn+\frac{n}{t}$

by using the maximal principle, which has a little difference from our result.
(2) We claim that (4.3) also holds for the system (4.1)-(4.2). Since

$\frac{\partial}{\partial\nu}\hat{S}_{t}=h_{t}\frac{\partial}{\partial\nu}\left(\frac{|\nabla
u|^{2}}{u}-\Delta u\right)=-2h_{t}\frac{1}{u^{2}}\mathbb{I}_{t}(\nabla
u,\nabla u)\leq 0,$

where the second equality is ensured by the proof of [4, Theorem 3.4], we
conclude that $\hat{S}_{t}$ for $c_{1}=2,c_{2}=1$ is also a submartingale,
which lead to the assertion announced above.

When the manifold $M$ is noncompact, we present the local version of the
gradient estimates under the Ricci flow.

######  Theorem 4.2.

Suppose $(M,g_{t})_{t\in[0,T]}$ is a complete solution of the Ricci flow
(1.2). Assume that $|{\rm Ric}_{t}|\leq k$ for some $k>0$ and all $(x,t)\in
B_{\rho,T}$, and a smooth positive function
$u:M\times[0,T]\rightarrow\mathbb{R}$ satisfies the heat equation (1.1). Then
for $\alpha>1$ and $t>0$,

$\displaystyle\left|\frac{\nabla u}{u}\right|^{2}\leq
2\left(\frac{1}{t}+\frac{4\pi^{2}(n+7)}{(4-\pi)^{2}\rho^{2}}+\frac{8k\pi}{(4-\pi)^{2}}\right)\left(4+\log\frac{\|u\|_{B_{\rho,T}}}{u}\right),$

and

$\frac{|\nabla u|^{2}}{u^{2}}-\alpha\frac{\Delta
u}{u}\leq\frac{2\alpha^{2}n}{t}+\frac{8\alpha^{2}n\pi^{2}}{(4-\pi)^{2}}\frac{n(1+{\alpha^{2}}{(\alpha-1)^{-1}})+3}{\rho^{2}}+\left(\frac{4+\pi}{4-\pi}\right)^{2}\alpha^{2}kn+\frac{\alpha^{3}kn}{\alpha-1}$

hold on $B_{\frac{\rho}{2},T}$.

###### Proof.

Under the Ricci flow, $\mathcal{R}_{t}\equiv 0$, then (2.6) holds for
$k_{1}+k_{2}=0$. Moreover, since ${\rm Ric}_{t}=-\partial_{t}g_{t}\leq k$, we
have

$\displaystyle(\Delta-2\partial_{t})\rho_{t}(x_{0},\cdot)(x)\leq\frac{n-1}{\rho_{t}(x_{0},x)}+k\rho_{t}(x_{0},x),$
(4.4)

which implies the first inequity. On the other hand, by the curvature
condition, we see that the inequity (3.8) holds for $k_{1}=k_{2}=k_{3}=k$ and
$k_{4}=0$. Thus, according to (4.4) and Lemma 3.2, we complete the proof. ∎

Acknowledgements The author thanks Professor Feng-Yu Wang for his guidance.
This work is supported in part by 985 Project, 973 Project.

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