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Big Corporates Pave the Path: Embracing AI and Cybersecurity for Success

(Bloomberg) -- What started as a promising year for emerging-market local debt is quickly souring as a strong dollar and a barrage of US tariffs have fund managers rethinking one of their main bets for developing assets. Most Read from BloombergNew York’s First ‘Passive House’ School ...

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Claims have emerged in which Musk seems to admit he did secretly enlist help in climbing the leaderboard for Path of Exile 2.

Elon Musk appears to admit he’s guilty of cheating in gaming scandal | Fortune

Musk's livestreams in which he plays a popular role-playing game have sparked some in the gaming community to speculate that he's not the gamer he claims to be.

Elon Musk faces gamers’ wrath over Path of Exile 2 streams

Achieving that level with a hardcore character is a genuinely impressive feat, but Path of Exile 2 players began calling his purported skills into question based on gameplay seen in the stream that is decidedly not what you'd expect from any high level player, much less one who holds one of ...

Elon Musk and Asmongold are beefing over who the best gamer really is, and no matter who wins, we lose | PC Gamer

Probably by paying other people to play for him, a common practice in games know as account boosting. At least that was one conclusion reached by other Path of Exile 2 players, including streamers Casual Kripp and Quin69TV.

Elon Musk And Asmongold Are Fighting After The Streamer Accused Him Of Being A Fake Gamer

Elon Musk is currently in the midst of a feud with Twitch streamer Asmongold, after he was accused of boosting his ‘Path of Exile 2’ account.

Elon Musk Vs. Asmongold—The Gaming Feud, Explained

GAMING News: Elon Musk is known as a top player in games like Diablo IV and Path of Exile 2. Some gamers suspect Musk hired skilled players or used exploits to boo

Elon Musk’s gaming stats questioned by top players: “He literally has no idea…” - Times of India

Trikon and SFT Protocol have announced a strategic collaboration aimed at transforming the Web3 ecosystem.

Trikon and SFT Protocol Forge Path for Next-Gen Web3 Gaming

Path

path

Keep your happy path flowing, not nesting Keep your happy path flowing, not nesting TL;DR: Arrange your code so the main logic flows along the left margin, handling edge cases early with guard clauses. TL;DR: Arrange your code so the main logic flows along the left margin, handling edge cases early with guard clauses. Problems 😔 Cognitive overhead
Readability
Excessive indentation
Maintainability
Control flow confusion
Stairs Code Cognitive overhead Readability Excessive indentation indentation Maintainability Control flow confusion Stairs Code Stairs Code Solutions 😃 Use early returns
Apply guard clauses
Handle errors first
Keep the main flow to the left
Minimize nesting depth Use early returns Apply guard clauses Handle errors first Keep the main flow to the left Minimize nesting depth Context 💬 When you write code with deeply nested conditional structures, you create "arrow code" or "pyramid of doom." arrow code This makes your program's primary flow hard to follow as it zigzags deeper into indentation levels. Your main logic (the "happy path") gets buried under layers of conditions, making the code harder to read, understand, and maintain. conditions This becomes even more problematic when dealing with internationalization and localization. Nested conditionals often create fragmented contexts for strings, making accurate translations difficult because translators lose the surrounding context needed for proper translation. Sample Code 📖 Wrong ❌ function processUserOrder(user, items) {
  if (user) {
    if (user.isActive()) {
      if (items.length > 0) {
        if (user.hasEnoughCredit()) {
          // The actual business logic is buried 4 levels deep
          let order = createOrder(user, items);
          notifyUser(user, 
            `Your order has been processed`);
          return order;
        } else {
          throw new Error("Insufficient credit");
        }
      } else {
        throw new Error("No items in cart");
      }
    } else {
      throw new Error("Your account is inactive");
    }
  } else {
    throw new Error("No user provided");
  }
} function processUserOrder(user, items) {
  if (user) {
    if (user.isActive()) {
      if (items.length > 0) {
        if (user.hasEnoughCredit()) {
          // The actual business logic is buried 4 levels deep
          let order = createOrder(user, items);
          notifyUser(user, 
            `Your order has been processed`);
          return order;
        } else {
          throw new Error("Insufficient credit");
        }
      } else {
        throw new Error("No items in cart");
      }
    } else {
      throw new Error("Your account is inactive");
    }
  } else {
    throw new Error("No user provided");
  }
} Right 👉 function processUserOrder(user, items) {
  if (!user) throw new Error("No user provided");
  if (!user.isActive()) throw new Error("Your account is inactive");
  if (items.length === 0) throw new Error("No items in cart");
  if (!user.hasEnoughCredit()) throw new Error("Insufficient credit");

  const order = createOrder(user, items);
  notifyUser(user,
    `Your order has been processed`);
  return order;
}

// This is even more readable

function assertValidOrder(user, items) {
  if (!user) throw new Error("No user provided");
  if (!user.isActive()) throw new Error("Your account is inactive");
  if (items.length === 0) throw new Error("No items in cart");
  if (!user.hasEnoughCredit()) throw new Error("Insufficient credit");
}

function processUserOrder(user, items) {
  assertValidOrder(user, items);
  const order = createOrder(user, items);
  notifyUser(user,
    `Your order has been processed`);
  return order;
} function processUserOrder(user, items) {
  if (!user) throw new Error("No user provided");
  if (!user.isActive()) throw new Error("Your account is inactive");
  if (items.length === 0) throw new Error("No items in cart");
  if (!user.hasEnoughCredit()) throw new Error("Insufficient credit");

  const order = createOrder(user, items);
  notifyUser(user,
    `Your order has been processed`);
  return order;
}

// This is even more readable

function assertValidOrder(user, items) {
  if (!user) throw new Error("No user provided");
  if (!user.isActive()) throw new Error("Your account is inactive");
  if (items.length === 0) throw new Error("No items in cart");
  if (!user.hasEnoughCredit()) throw new Error("Insufficient credit");
}

function processUserOrder(user, items) {
  assertValidOrder(user, items);
  const order = createOrder(user, items);
  notifyUser(user,
    `Your order has been processed`);
  return order;
} Detection 🔍 Semi-Automatic Semi-Automatic You can detect this smell by looking for multiple indentation levels (more than 2 or 3). You can also analyse ASTs with advanced linters. Tags 🏷️ IFs IFs Level 🔋 Beginner Beginner Why the Bijection Is Important 🗺️ When you write code with deep nesting, you break the clean Bijection between the logical flow of your business rules and their representation in code. Bijection The real-world business process likely follows a series of validations followed by a main action, but deeply nested code obscures this natural sequence. This one-to-one correspondence breaks down because the primary operation (what the function is supposed to do) gets buried deep in indentation layers. The logical sequence of validations isn't separated from the main action. By keeping your happy path to the left, you create a natural bijection between the actual process flow and the code structure, making it easier to reason about and modify in the future. AI Generation 🤖 AI code generators often create nested conditional structures, especially when generating code from prompts that don't explicitly request early returns or guard clauses. Many AI systems mimic common patterns they observe in training data, where deeply nested conditions are unfortunately prevalent. AI Detection 🥃 Most AI code assistants can identify and fix this code smell with proper instructions. If you ask an AI to refactor code to "use early returns" or "apply guard clauses" or "keep the happy path to the left," it can typically transform nested conditionals into flatter structures. You can also prompt the AI to "reduce nesting in this function" or "refactor this code to avoid deep indentation," and set it as a meta-prompt following your style preferences. Try Them! 🛠 Remember: AI Assistants make lots of mistakes Remember: AI Assistants make lots of mistakes Suggested Prompt: Remove the deep nesting Suggested Prompt: Remove the deep nesting Without Proper Instructions

With Specific Instructions



ChatGPT

ChatGPT



Claude

Claude



Perplexity

Perplexity



Copilot

Copilot



Gemini

Gemini



DeepSeek

DeepSeek



Meta AI

Meta AI



Qwen

Qwen Without Proper Instructions

With Specific Instructions



ChatGPT

ChatGPT



Claude

Claude



Perplexity

Perplexity



Copilot

Copilot



Gemini

Gemini



DeepSeek

DeepSeek



Meta AI

Meta AI



Qwen

Qwen Without Proper Instructions

With Specific Instructions Without Proper Instructions Without Proper Instructions With Specific Instructions With Specific Instructions ChatGPT

ChatGPT ChatGPT ChatGPT ChatGPT ChatGPT ChatGPT ChatGPT Claude

Claude Claude Claude Claude Claude Claude Claude Perplexity

Perplexity Perplexity Perplexity Perplexity Perplexity Perplexity Perplexity Copilot

Copilot Copilot Copilot Copilot Copilot Copilot Copilot Gemini

Gemini Gemini Gemini Gemini Gemini Gemini Gemini DeepSeek

DeepSeek DeepSeek DeepSeek DeepSeek DeepSeek DeepSeek DeepSeek Meta AI

Meta AI Meta AI Meta AI Meta AI Meta AI Meta AI Meta AI Qwen

Qwen Qwen Qwen Qwen Qwen Qwen Qwen Conclusion 🏁 Keep your happy path to the left by using early returns and guard clauses, you will create more readable, maintainable code. You communicate business logic more clearly, reduce cognitive load for other developers (including your future self), and create more resilient code to change. Remember to handle the special cases early, and let your main logic flow naturally along the left margin. Your colleagues (and future you) will thank you. Relations 👩‍❤️‍💋‍👨 Code Smell 102 - Arrow Code Code Smell 102 - Arrow Code https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxiv?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxiv?embedable=true https://maximilianocontieri.com/code-smell-294-implicit-return?embedable=true https://maximilianocontieri.com/code-smell-294-implicit-return?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-i-xqz3evd?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-i-xqz3evd?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxvii?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxvii?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxiii?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxiii?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxi?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxi?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxvii https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-xxxvii Disclaimer 📘 Code Smells are my opinion. opinion Credits 🙏 Photo by Alexander Hipp on Unsplash Alexander Hipp Unsplash A function should follow the "arrow shape" of reading naturally from top to bottom, not wander into deeper nesting like a poorly designed maze. A function should follow the "arrow shape" of reading naturally from top to bottom, not wander into deeper nesting like a poorly designed maze. Venkat Subramaniam Software Engineering Great Quotes Software Engineering Great Quotes This article is part of the CodeSmell Series. https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-i-xqz3evd?embedable=true https://hackernoon.com/how-to-find-the-stinky-parts-of-your-code-part-i-xqz3evd?embedable=true

Keep Your Happy Path Flowing, Not Nesting

So, you want to become better than 99% of programmers. But you are doing the exact same things that 99% of programmers are already doing — watching video tutorials, doing online courses, and solving Leetcode problems. Everybody is doing that. Today, we are going to discuss how, if we are in the field of coding and programming, we can go into the top 1% of our field. How we can stand out from the crowd and move forward. Now, all the things we are going to discuss today will be fairly simple; there is no need for us to make any major sacrifices. But yes, realistically speaking, none of these are things that if we do for one day, we will reach the top 1%. These are things on which we will have to invest conscious effort and consciously invest time, and gradually, if we keep following all these things over time, then definitely we will reach the top 1% in our field. “The secret of getting ahead is getting started.” — Mark Twain “The secret of getting ahead is getting started.” — Mark Twain “The secret of getting ahead is getting started.” Doing what everyone else is doing is the very definition of how to become average. And there’s nothing wrong with that. But if you want to get ahead of others, you have to be willing to do what others are not willing to do. Here are such things you can do right now. Enjoy the Process Now, the very first thing we can do among these is to enjoy the process. So basically, the thing we can pick from here is the mindset. About our work — the more meaningful we understand that whatever work we are doing, whatever things we are doing, whatever skill we are trying to build feels to us as meaningful, the more we enjoy that thing. “Pleasure in the job puts perfection in the work.” — Aristotle “Pleasure in the job puts perfection in the work.” — Aristotle “Pleasure in the job puts perfection in the work.” The more that thing brings happiness for us. Therefore, if we want to become the top 1% of coders, then we will have to genuinely enjoy the process of coding. In every field, the top % of people are those who genuinely enjoy their work more than other people. Therefore, it is important for us to consciously enjoy it as well. The Right Mindset for Growth Now, here, “enjoying” does not mean at all that if we are watching a movie or going out with friends or eating good food, we will start enjoying coding to that extent. Definitely, it is such a thing in which we will be putting in more effort; but at the end, doing this does not mean that if we code merely for mental peace or a sense of satisfaction, then my job will get done — this is absolutely the wrong approach. “Your mind is a powerful thing. When you fill it with positive thoughts, your life will start to change.” — Unknown “Your mind is a powerful thing. When you fill it with positive thoughts, your life will start to change.” — Unknown “Your mind is a powerful thing. When you fill it with positive thoughts, your life will start to change.” Because after a week, that thing will slip from our mind, and then we will not feel like doing it. So, we will have to consciously enjoy our work, find meaning in it so that that thing actually makes us happy — so that the whole process of coding and programming starts giving us happiness. Master Debugging Skills Most people think that programmers code all day. In reality, programmers spend most of their time debugging code. Breaking things and fixing them is an essential skill to become a skilled programmer. “It’s fine to celebrate success, but it is more important to heed the lessons of failure.” — Bill Gates “It’s fine to celebrate success, but it is more important to heed the lessons of failure.” — Bill Gates “It’s fine to celebrate success, but it is more important to heed the lessons of failure.” Yet, beginners don’t spend any time learning the right way to debug code. To get started with learning debugging, you can do this course called Software Debugging on Udacity. Software Debugging on Udacity Software Debugging on Udacity Software Debugging on Udacity Software Debugging on Udacity . Look Out for Opportunities The thing that we can do is to look out for opportunities — opportunities that give us the chance to improve our existing skills and opportunities that give us the chance to learn new skills. We can also call this being curious. We can also call this experimentation, but basically, we have to keep looking for such opportunities where we get a chance to do something within tech. “In the middle of every difficulty lies opportunity.” — Albert Einstein “In the middle of every difficulty lies opportunity.” — Albert Einstein “In the middle of every difficulty lies opportunity.” Do not ignore any small opportunity; in fact, it is about seeking out new opportunities, finding them — because in tech, many times you will see successful people who started with a very small, very random thing, and then that thing kick-started their career — they brought a big change in their career. So, we should not be lazy, we have to actively go and seek out opportunities. Quality Over Quantity When starting out, many beginners watch the longest possible programming tutorials, learn as many programming languages as they can, and build as many projects as possible. In other words, quantity is often assumed to be all that matters. “It is quality rather than quantity that matters.” — Seneca “It is quality rather than quantity that matters.” — Seneca “It is quality rather than quantity that matters.” But in programming, quality matters much more than quantity. One great project on your resume can easily give you an edge over someone who has 10 basic projects on their resume. quality matters much more than quantity Invest Time in Your Skills Time does not mean that as our age increases, we will become an inventive coder. The time here means that the more hours we spend on this skill, the more inventive we will become. Those hours you can spend either after you get a job or you can start spending them during your college life. But basically, this is a skill, and for learning this skill, the sooner and the more hours you invest, the better you will become in that skill. “Time is the most valuable thing a man can spend.” — Theophrastus “Time is the most valuable thing a man can spend.” — Theophrastus “Time is the most valuable thing a man can spend.” The first of these is that we invest time in making projects using our skills because whenever we invest our time in practical learning, it will bring us more fruitful outputs. The second thing is that we invest our time in reading documentation, learning, and looking at the work of other people. The Importance of Reading Code Think about this: If a programmer writes 10,000 lines of code over two years, how many lines of code do you think they have read during the same period? It would easily be 100,000 lines of code, which is 10 times what they have written. Yet, new programmers spend no time reading code written by experienced programmers. Go to GitHub, and spend 30 minutes trying to understand someone else’s code every day. Think about this: If a programmer writes 10,000 lines of code over two years, how many lines of code do you think they have read during the same period? It would easily be 100,000 lines of code, which is 10 times what they have written. Yet, new programmers spend no time reading code written by experienced programmers. Go to GitHub, and spend 30 minutes trying to understand someone else’s code every day. GitHub, 30 minutes trying to understand someone else’s code every day “A reader lives a thousand lives before he dies. The man who never reads lives only one.” — George R.R. Martin “A reader lives a thousand lives before he dies. The man who never reads lives only one.” — George R.R. Martin “A reader lives a thousand lives before he dies. The man who never reads lives only one.” Many times in tech, we know how to write our own code, and we know how to write our logic, but it is also important in tech that we read and understand other people’s code. In fact, if we go and read the code, documentation, or random articles, blogs, etc., of those industry professionals who are already working and coding professionally, then that gives us inventiveness. So, as a programmer, just writing code is not enough; we should also establish good reading habits. Because many times, when we go to work in a company, we will have to learn things by reading documentation — there, we will not be taught in a way that makes us sit like in school. Collaborating With Other Programmers & Networking The biggest disadvantage beginners face is that often, they don’t get the opportunity to collaborate with experienced programmers. As a result, they never learn what it’s like to collaborate with other engineers on a team. To solve this problem, you can start contributing to open-source projects. you can start contributing to open-source projects “Alone we can do so little; together we can do so much.” — Helen Keller “Alone we can do so little; together we can do so much.” — Helen Keller “Alone we can do so little; together we can do so much.” If you are working in a company, look out for people who can help you grow and whom you can help grow. And the type of networking we can do is with the seniors in our industry. Many times, seniors have already made those mistakes, so there are some common mistakes that they prevent us from making. Many times, they recommend good resources and good opportunities to us. And many times, they give us the right career advice which we can follow. So, in this way, when we network with some good mentors in our field, it is not that they will only help us in college or help with referrals for the first job; we will get help from these people for a lifetime as long as we are working in our tech industry. So, in this way, making good connections with the people in our industry and networking well, is important to grow in this field. Learning From the Best To become a top 1% coder, you need to know what a top 1% coder looks like. But where can you even find a top 1% coder? At conferences and meetups. To find meetups happening near you, you can use meetup.com. The First Step: Thinking Like a Programmer. But before you can do any of this, you need to know how to think like a programmer. top 1% coder top 1% coder conferences and meetups meetup.com meetup.com how to think like a programmer “If I have seen further, it is by standing on the shoulders of giants.” — Isaac Newton “If I have seen further, it is by standing on the shoulders of giants.” — Isaac Newton “If I have seen further, it is by standing on the shoulders of giants.” Final Thoughts on Becoming a Top 1% Programmer Now, these are some things that can benefit the maximum number of us to reach the top 1%, and to follow these is not absolutely necessary. It may be that we are already a working professional, working in a company; even then, if we follow these things, we will see results in the end. Now, these are the kinds of things that, if we start doing them, we will not see results overnight, but I can assure that if you try to include these things in your daily routine with conscious effort, taking small steps — if we do every day, then after 30 days, after one month, we will definitely start seeing some changes. “We are what we repeatedly do. Excellence, then, is not an act, but a habit.” — Aristotle “We are what we repeatedly do. Excellence, then, is not an act, but a habit.” — Aristotle “We are what we repeatedly do. Excellence, then, is not an act, but a habit.” And after 6 months, we will have a strong belief that by definitely practicing these things in our career and professional life, by putting in conscious effort every day, we can move towards the top 1% in our field. So, I hope to see a lot of you becoming the top 1% in this field. That’s all for today; see you in the next story. Till then, keep learning and keep exploring. Let’s grow, learn, and build amazing things together! Don’t forget to like your heart content, save it to your list, and follow me. Stay connected with me on my other platforms:LinkedIn | dev.to| Bluesky | HackerNoon Stay connected with me on my other platforms:LinkedIn LinkedIn LinkedIn dev.to dev.to dev.to Bluesky | HackerNoon Bluesky Bluesky HackerNoon HackerNoon

How to Become a Top 1% Programmer: The Path You Have to Take

Welcome to HackerNoon’s Hacking Tech Marketing template. It is designed to you in showcasing your (or your company’s) recent triumphs and successes in the realm of Technology Marketing. Welcome to HackerNoon’s Hacking Tech Marketing template. It is designed to you in showcasing your (or your company’s) recent triumphs and successes in the realm of Technology Marketing. 1. Introduction 1. Introduction Ranking #1 on Google won’t guarantee subscription growth—what matters is what happens after users land on your site. Understanding their journey is the missing link to increasing conversions. Ranking #1 on Google won’t guarantee subscription growth—what matters is what happens after users land on your site. Understanding their journey is the missing link to increasing conversions. In today's competitive digital landscape, subscription-based businesses face unique challenges in attracting and retaining customers. Traditional SEO strategies often emphasize keyword optimization and ranking improvements. However, understanding the user journey—the path a visitor takes from initial awareness to subscription—is equally crucial for driving conversions and fostering long-term customer relationships. Why Focus on User Journey Analysis? Why Focus on User Journey Analysis? Targeted Traffic: SEO is not just about increasing website traffic; it's about attracting visitors who are more likely to convert into subscribers.
Behavioral Insights: By analyzing user behavior, businesses can identify friction points that hinder conversions and address them effectively.
Enhanced Engagement: Tailoring content and website structure to align with user needs at each stage of their journey can lead to higher engagement and satisfaction.

In this article, we'll explore how to align SEO strategies with user journeys using Google Analytics 4 (GA4) Path Exploration, optimize conversion paths, and enhance subscription growth through data-driven insights. Targeted Traffic: SEO is not just about increasing website traffic; it's about attracting visitors who are more likely to convert into subscribers.
Behavioral Insights: By analyzing user behavior, businesses can identify friction points that hinder conversions and address them effectively.
Enhanced Engagement: Tailoring content and website structure to align with user needs at each stage of their journey can lead to higher engagement and satisfaction. Targeted Traffic: SEO is not just about increasing website traffic; it's about attracting visitors who are more likely to convert into subscribers. Targeted Traffic: Behavioral Insights: By analyzing user behavior, businesses can identify friction points that hinder conversions and address them effectively. Behavioral Insights: Enhanced Engagement: Tailoring content and website structure to align with user needs at each stage of their journey can lead to higher engagement and satisfaction. Enhanced Engagement: In this article, we'll explore how to align SEO strategies with user journeys using Google Analytics 4 (GA4) Path Exploration, optimize conversion paths, and enhance subscription growth through data-driven insights. 2. Understanding the User Journey for Subscription-Based SEO2.1 What is the User Journey? 2. Understanding the User Journey for Subscription-Based SEO2.1 What is the User Journey? 2.1 What is the User Journey? 2.1 What is the User Journey? The user journey represents the path a visitor takes from discovering a website to subscribing. For subscription-based models, this typically consists of four key stages: Awareness Stage: Users discover the website through organic search, social media, or referrals.
Consideration Stage: Users engage with content, read case studies, or compare subscription options.
Decision Stage: Users are on key conversion pages (pricing, signup) and may subscribe or abandon the process.
Retention Stage: Post-subscription, users may cancel, upgrade, or engage with additional content. Awareness Stage: Users discover the website through organic search, social media, or referrals. Awareness Stage: Consideration Stage: Users engage with content, read case studies, or compare subscription options. Consideration Stage: Decision Stage: Users are on key conversion pages (pricing, signup) and may subscribe or abandon the process. Decision Stage: Retention Stage: Post-subscription, users may cancel, upgrade, or engage with additional content. Retention Stage: Neglecting any of these stages can lead to higher drop-offs and lower conversion rates. 2.2 Why User Journey Matters More in Subscription Businesses Compared to E-commerce 2.2 Why User Journey Matters More in Subscription Businesses Compared to E-commerce In industries like e-commerce, the user journey is often shorter and more transactional. A user searches for a product using a high-intent keyword, lands on a category or product page, and may make a purchase after evaluating the options. The journey is relatively direct, with clear signals of purchase intent. However, in subscription-based models, particularly for digital magazines, SaaS, or streaming platforms, the process is fundamentally different. Users typically arrive at the site through informational search queries, exploring articles, industry trends, or educational content. Unlike e-commerce, where transactional intent is immediate, a subscription-based journey often involves multiple touchpoints before a user decides to subscribe. subscription-based models For example, a reader searching for “latest AI advancements” might land on an article in a digital magazine. If the publication effectively guides them to related content and demonstrates value, the user may choose to subscribe after engaging with several pieces of content. This highlights why tracking user interactions across multiple pages is essential for improving subscription conversions. “latest AI advancements” 2.3 How SEO Impacts Different Stages of the User Journey 2.3 How SEO Impacts Different Stages of the User Journey To optimize subscription conversions, businesses need to align their SEO strategy with each journey stage. Below is a framework for doing so: User Journey Stage

SEO Focus

Key Metrics to Track



Awareness

Optimize blog content, TOFU keywords, schema markup

Impressions, CTR, Organic traffic



Consideration

Improve internal linking, E-A-T content, UX enhancements

Engagement rate, Time on page, Returning users



Decision

Optimize pricing pages, add persuasive content, CRO for SEO

Subscription conversions, Form completions



Retention

Retargeting SEO, personalized content recommendations

Churn rate, Returning visits, Upsell conversions User Journey Stage

SEO Focus

Key Metrics to Track



Awareness

Optimize blog content, TOFU keywords, schema markup

Impressions, CTR, Organic traffic



Consideration

Improve internal linking, E-A-T content, UX enhancements

Engagement rate, Time on page, Returning users



Decision

Optimize pricing pages, add persuasive content, CRO for SEO

Subscription conversions, Form completions



Retention

Retargeting SEO, personalized content recommendations

Churn rate, Returning visits, Upsell conversions User Journey Stage

SEO Focus

Key Metrics to Track User Journey Stage User Journey Stage SEO Focus SEO Focus Key Metrics to Track Key Metrics to Track Awareness

Optimize blog content, TOFU keywords, schema markup

Impressions, CTR, Organic traffic Awareness Awareness Optimize blog content, TOFU keywords, schema markup Optimize blog content, TOFU keywords, schema markup Impressions, CTR, Organic traffic Impressions, CTR, Organic traffic Consideration

Improve internal linking, E-A-T content, UX enhancements

Engagement rate, Time on page, Returning users Consideration Consideration Improve internal linking, E-A-T content, UX enhancements Improve internal linking, E-A-T content, UX enhancements Engagement rate, Time on page, Returning users Engagement rate, Time on page, Returning users Decision

Optimize pricing pages, add persuasive content, CRO for SEO

Subscription conversions, Form completions Decision Decision Optimize pricing pages, add persuasive content, CRO for SEO Optimize pricing pages, add persuasive content, CRO for SEO Subscription conversions, Form completions Subscription conversions, Form completions Retention

Retargeting SEO, personalized content recommendations

Churn rate, Returning visits, Upsell conversions Retention Retention Retargeting SEO, personalized content recommendations Retargeting SEO, personalized content recommendations Churn rate, Returning visits, Upsell conversions Churn rate, Returning visits, Upsell conversions Common SEO Mistakes in Subscription-Based Businesses: ❌ Over-focusing on top-of-funnel (TOFU) keywords without guiding users to conversion. ❌ Ignoring UX and technical SEO for subscription pages. ❌ Failing to track user behavior beyond search rankings. 3. GA4 Path Exploration: A Game-Changer for Subscription SEO 3. GA4 Path Exploration: A Game-Changer for Subscription SEO 3.1 What is GA4 Path Exploration? 3.1 What is GA4 Path Exploration? GA4’s Path Exploration feature helps track how users navigate through a website before converting. It enables businesses to: Identify high-exit pages that cause drop-offs.
Track the most common paths to subscription pages.
Optimize internal linking and CTA placements. Identify high-exit pages that cause drop-offs. Identify high-exit pages that cause drop-offs. Track the most common paths to subscription pages. Track the most common paths to subscription pages. Optimize internal linking and CTA placements. Optimize internal linking and CTA placements. 3.2 Step-by-Step Guide: Setting Up GA4 Path Exploration 3.2 Step-by-Step Guide: Setting Up GA4 Path Exploration Step 1: Open Path Exploration in GA4 Step 1: Open Path Exploration in GA4 Navigate to Google Analytics 4, click on Explore, and select Path Exploration. This section provides a visual representation of how users navigate through your website. Step 2: Start a New Exploration Click Start Over to reset any previous sessions and begin fresh analysis. Step 3: Select an Ending Point Click Ending Point and define what success looks like, such as a Subscription Confirmation Page or a Sign-Up Completion Event. Step 4: Choose the Key Event Click Load More or use the search bar to find and select your end goal event (e.g., “Subscription”). 📌 Note: GA4 does not automatically track subscription events. You must set them up using Google Tag Manager to measure completed subscriptions accurately. Step 5: Segment Users by Behavior Apply segments to compare different user groups: New vs. Returning Users: See how first-time visitors behave compared to repeat users.
Organic vs. Paid Traffic: Identify which acquisition channels drive the most subscriptions. New vs. Returning Users: See how first-time visitors behave compared to repeat users. New vs. Returning Users Organic vs. Paid Traffic: Identify which acquisition channels drive the most subscriptions. Organic vs. Paid Traffic Step 6: Analyze User Paths & Drop-Off Points Click on nodes to trace user behavior up to 10 steps backward: Identify high drop-off points where users abandon their journey.
Find high-engagement pages that contribute to successful conversions.
Discover the most effective pathways users take to subscribe. Identify high drop-off points where users abandon their journey. Find high-engagement pages that contribute to successful conversions. Discover the most effective pathways users take to subscribe. By following these steps, businesses can uncover data-driven insights to improve SEO, refine content strategy, and optimize conversion paths for subscription growth. 4. Aligning SEO Strategy with GA4 Path Exploration Insights Understanding how users navigate your site and identifying where they drop off is crucial in refining your SEO strategy for better conversions. GA4 Path Exploration helps uncover key friction points that prevent users from moving through the funnel, allowing businesses to implement targeted SEO improvements. Below are some examples of how specific SEO actions can be aligned with different user journey stages based on GA4 insights. Addressing Drop-offs from Blog to Pricing Page Addressing Drop-offs from Blog to Pricing Page If GA4 data shows that users frequently drop off between a blog page and a pricing page, it suggests a lack of clear navigation or compelling CTAs. Businesses should strategically add internal links within high-traffic blog posts and ensure that CTAs are prominent and persuasive. This approach helps seamlessly transition readers from informational content to subscription-focused pages, driving more traffic to decision-stage content. Reducing High Bounce Rates on Signup Pages Reducing High Bounce Rates on Signup Pages A high bounce rate on a signup page often signals usability issues such as slow load times, an unclear value proposition, or trust concerns. To combat this, businesses should enhance page speed, improve the overall UX design, and add credibility elements such as testimonials, security badges, and money-back guarantees. These adjustments can reduce friction and significantly increase conversion rates. Leveraging Returning Users for Higher Conversions Leveraging Returning Users for Higher Conversions If GA4 indicates that returning users are more likely to convert, it presents an opportunity to implement a retargeting SEO strategy. Businesses should create personalized content experiences, use email marketing, and implement remarketing ads to nurture these users toward subscription. Offering exclusive content previews or limited-time subscription discounts can also encourage returning visitors to commit. 5. Actionable SEO Strategies to Drive Subscription Growth 5. Actionable SEO Strategies to Drive Subscription Growth ✔️ **Set up GA4 Path Exploration & track subscription events \ Ensure that key subscription interactions—such as free trials, sign-ups, and churn events—are correctly tracked in GA4 to gain a full picture of the user journey. Ensure that key subscription interactions—such as free trials, sign-ups, and churn events—are correctly tracked in GA4 to gain a full picture of the user journey. ✔️ **Identify drop-off points & high-engagement pages \ Use GA4 Path Exploration to analyze where users leave your site and which content drives the most engagement, then optimize accordingly. Use GA4 Path Exploration to analyze where users leave your site and which content drives the most engagement, then optimize accordingly. ✔️ **Optimize content & internal linking based on user journey stage \ Enhance content structure by linking high-traffic pages to conversion-focused pages and aligning topics with each user journey stage. Enhance content structure by linking high-traffic pages to conversion-focused pages and aligning topics with each user journey stage. ✔️ **Use GA4 + GSC data to prioritize high-converting SEO opportunities \ Merge GA4 insights with Google Search Console data to identify top-performing keywords and prioritize SEO strategies that lead to subscriptions. Merge GA4 insights with Google Search Console data to identify top-performing keywords and prioritize SEO strategies that lead to subscriptions. ✔️ **Improve technical SEO for retention-stage users (e.g., site speed, trust signals) \ Ensure a seamless user experience with fast-loading pages, mobile optimization, and strong trust-building elements to retain subscribers. Ensure a seamless user experience with fast-loading pages, mobile optimization, and strong trust-building elements to retain subscribers. 6. Conclusion 6. Conclusion The shift from traditional transactional SEO to user journey-driven SEO is critical for subscription-based businesses. Unlike e-commerce models, where users often make one-time purchases with clear transactional intent, subscription models require ongoing engagement, trust-building, and multiple interactions before conversion. This makes understanding and optimizing the user journey a core pillar of SEO success in this industry. Key Takeaways Key Takeaways SEO for subscriptions is more than rankings—it’s about guiding users through multiple touchpoints toward conversion and retention.
GA4 Path Exploration reveals critical insights—helping businesses track user behavior, identify friction points, and optimize conversion pathways.
Combining GA4 with Google Search Console & BigQuery allows businesses to make data-backed SEO decisions, focusing on high-converting keywords and content strategies.
Retention matters as much as acquisition—technical SEO, content personalization, and UX improvements are essential for reducing churn and increasing lifetime value (LTV). SEO for subscriptions is more than rankings—it’s about guiding users through multiple touchpoints toward conversion and retention. GA4 Path Exploration reveals critical insights—helping businesses track user behavior, identify friction points, and optimize conversion pathways. Combining GA4 with Google Search Console & BigQuery allows businesses to make data-backed SEO decisions, focusing on high-converting keywords and content strategies. Retention matters as much as acquisition—technical SEO, content personalization, and UX improvements are essential for reducing churn and increasing lifetime value (LTV). Final Thoughts Final Thoughts Subscription-based businesses that embrace a data-driven SEO approach will have a competitive edge in converting and retaining subscribers. By integrating GA4 insights with a strategic SEO plan, businesses can create a seamless, optimized user journey that not only attracts users but keeps them engaged long-term. The next step? Implementing GA4 Path Exploration and continuously refining your SEO strategy based on user behavior insights. 📢 How are you using GA4 insights to optimize your subscription-based SEO? Share your biggest challenges and wins in the comments, or connect with me for more in-depth discussions! 🚀 How are you using GA4 insights to optimize your subscription-based SEO? Share your biggest challenges and wins in the comments, or connect with me for more in-depth discussions! Would you like to take a stab at answering some of these questions? The link for the template is HERE, just start writing! Would you like to take a stab at answering some of these questions? The link for the template is HERE, just start writing! Would you like to take a stab at answering some of these questions? The link for the template is HERE, just start writing! HERE

Why GA4 Path Exploration is a Game-Changer for Subscription-Based SEO

How can blockchain support confidential transactions while meeting regulatory demands? COTI launches its mainnet as a scalable Layer 2 to support confidential operations in Web3. COTI COTI announces the launch of its mainnet after more than two years of research and development. The network operates as a Layer 2 solution for confidential transactions and aims to serve as a bridge between privacy and compliance in the digital ecosystem. The platform utilizes an implementation of Garbled Circuits developed in collaboration with Soda Labs. This system is designed to offer significant speed improvements over existing solutions while providing a Privacy-on-Demand service across multiple blockchain networks. Addressing the Privacy Gap in Web3 The absence of viable privacy layers has limited institutional participation in areas such as Real World Assets (RWAs). This gap has prevented many institutions from protecting sensitive data and meeting compliance requirements. COTI's new mainnet is designed to enable businesses to secure proprietary data and allow individuals to access Web3 services without disclosing personal information. The system also supports AI agents to operate as independent entities, ensuring that both data privacy and operational compliance are maintained across various use cases. Technical Framework and Ecosystem Integration COTI's mainnet employs a Layer 2 architecture that records transactions on-chain while supporting high-speed processing. The use of Garbled Circuits enhances the speed of confidential transactions by up to 3,000 times compared to traditional methods. The network is structured to support Privacy-on-Demand, providing confidentiality while meeting compliance standards. A range of ecosystem partners has already integrated COTI’s privacy layer. Partners include Bancor with its Carbon DeFi platform, Band Protocol, MyEtherWallet (MEW), and PriveX. In addition, COTI is extending its operations to government entities and CBDC initiatives, including a pilot project with the Bank of Israel involving the Digital Shekel. Further expansion announcements are expected in the near future. Transition and User Incentives To support the mainnet launch, the COTI Foundation will distribute 100 $COTI tokens to tens of thousands of wallets. Users who add COTI’s Layer 2 network to MetaMask will be eligible for a token claim. The release also includes a new Explorer and Bridge to facilitate the upgrade from COTI V1 to V2, ensuring a smooth transition for current users. Final Thoughts The launch of COTI's mainnet signals a shift in how confidential transactions are managed in Web3. By providing a scalable and compliant solution, COTI positions its network as an enabler of secure and private blockchain operations. The move may pave the way for increased adoption in sectors that require both confidentiality and adherence to regulatory standards, potentially reshaping the landscape for digital financial services. Don’t forget to like and share the story! Vested Interest Disclosure: This author is an independent contributor publishing via our business blogging program. HackerNoon has reviewed the report for quality, but the claims herein belong to the author. #DYO Vested Interest Disclosure: This author is an independent contributor publishing via our business blogging program. HackerNoon has reviewed the report for quality, but the claims herein belong to the author. #DYO Vested Interest Disclosure: business blogging program business blogging program

Can Web3 Achieve Confidentiality and Compliance? COTI’s Mainnet Provides a Path

¿Web3 ukax Confidencialidad ukat Cumplimiento ukanak phuqhaspati? COTI’s Mainnet ukax mä Thakhi churaraki

Může Web3 dosáhnout důvěrnosti a souladu? Hlavní síť COTI poskytuje cestu

¿Puede la Web3 lograr confidencialidad y cumplimiento normativo? La red principal de COTI ofrece una solución.

Bisakah Web3 Mencapai Kerahasiaan dan Kepatuhan? Mainnet COTI Menyediakan Jalan

Web3は機密性とコンプライアンスを実現できるか？COTIのメインネットが道を開く

Ar „Web3“ gali užtikrinti konfidencialumą ir atitiktį? COTI pagrindinis tinklas suteikia kelią

Kan Web3 vertrouwelijkheid en naleving bereiken? Het hoofdnetwerk van COTI biedt een pad

Ko Web3 Inogona Kuwana Kuvanzika uye Kutevedzera? COTI's Mainnet Inopa Nzira

Може ли Веб3 да постигне поверљивост и усклађеност? ЦОТИ Маиннет пружа пут

Web3 ரகசியத்தன்மை மற்றும் இணக்கத்தை அடைய முடியுமா? COTI இன் Mainnet ஒரு பாதையை வழங்குகிறது.

Оё Web3 метавонад ба махфият ва мувофиқат ноил шавад? Шабакаи асосии COTI роҳро таъмин мекунад

Xana Web3 Yi Nga Fikelela Xihundla Na Ku Landzelela milawu? COTI’s Mainnet Yi Nyika Ndlela

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization In this section we present the possible generalisation of Assumptions A-1 and A-2 as well as the exit-time result for this more general process. In this paper we did not talk in details about the problem of existence and uniqueness of the self-interacting diffusion. While in the Lipschitz case (Assumption A-1) these questions are standard, locally Lipschitzness of the potentials add some complication. In [AdMKT23], the authors provide a proof for existence and uniqueness of self-interacting diffusion under the following assumptions that we take as a baseline: Assumptions A-1′. It is worth noting that Assumption A-1′ .ii is a combination of locally Lipschitz and polynomial growth conditions. Assumption A-1′ .iii has been introduced in [AdMKT23] to control the Lyapunov functional of the diffusion process to ensure that there is no explosion within a finite time. It is important to mention that, if required, this assumption can be replaced by another one that ensures the process’s global existence and uniqueness without any impact on the exit-time result. We can also relax Assumptions A-2. Indeed, the domain G does not necessarily have to be bounded, but it is necessary to add another assumption on the level sets instead. Consider: Assumptions A-2′. Under these assumptions we can obtain Kramers’ type law and the exit location result. Acknowledgments: This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). References [AdMKT23] Ashot Aleksian, Pierre del Moral, Aline Kurtzmann, and Julian Tugaut. Self-interacting diffusions: long-time behaviour and exitproblem in the convex case. preprint, March 2023. [BLR02] Michel Benaïm, Michel Ledoux, and Olivier Raimond. Selfinteracting diffusions. Probab. Theory Related Fields, 122(1):1–41, 2002. [BR03] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. II. Convergence in law. Ann. Inst. H. Poincaré Probab. Statist., 39(6):1043–1055, 2003. [BR05] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. III. Symmetric interactions. Ann. Probab., 33(5):1717–1759, 2005. [CGM08] P. Cattiaux, A. Guillin, and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1-2):19–40, 2008. [DR92] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields, 92(3):337–349, 1992. [DZ10] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. [FW12] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, third edition, 2012. Translated from the 1979 Russian original by Joseph Szücs. [HIP08] Samuel Herrmann, Peter Imkeller, and Dierk Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008. [KK12] Victor Kleptsyn and Aline Kurtzmann. Ergodicity of selfattracting motion. Electron. J. Probab., 17:no. 50, 37, 2012. [Kur10] Aline Kurtzmann. The ODE method for some self-interacting diffusions on R d . Ann. Inst. Henri Poincaré Probab. Stat., 46(3):618– 643, 2010. [NRW87] J. R. Norris, L. C. G. Rogers, and David Williams. Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields, 74(2):271–287, 1987. [RW00] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition. [TTV12] Pierre Tarrès, Bálint Tóth, and Benedek Valkó. Diffusivity bounds for 1D Brownian polymers. Ann. Probab., 40(2):695–713, 2012. [Tug12] Julian Tugaut. Exit problem of McKean-Vlasov diffusions in convex landscapes. Electron. J. Probab., 17:no. 76, 26, 2012. [Tug16] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions. Electron. Commun. Probab., 21:Paper No. 11, 7, 2016. [Tug18] Julian Tugaut. Exit-problem of McKean-Vlasov diffusions in double-well landscape. J. Theoret. Probab., 31(2):1013–1023, 2018. [Tug19] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions in double-wells landscape. ALEA Lat. Am. J. Probab. Math. Stat., 16(1):389–398, 2019. [VF70] A. D. Ventcel’ and M. I. Freıdlin. Small random perturbations of dynamical systems. Uspehi Mat. Nauk, 25(1 (151)):3–55, 1970. [Vil09] Cédric Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 4 Generalization In this section we present the possible generalisation of Assumptions A-1 and A-2 as well as the exit-time result for this more general process. In this paper we did not talk in details about the problem of existence and uniqueness of the self-interacting diffusion. While in the Lipschitz case (Assumption A-1) these questions are standard, locally Lipschitzness of the potentials add some complication. In [AdMKT23], the authors provide a proof for existence and uniqueness of self-interacting diffusion under the following assumptions that we take as a baseline: Assumptions A-1′. It is worth noting that Assumption A-1′ .ii is a combination of locally Lipschitz and polynomial growth conditions. Assumption A-1′ .iii has been introduced in [AdMKT23] to control the Lyapunov functional of the diffusion process to ensure that there is no explosion within a finite time. It is important to mention that, if required, this assumption can be replaced by another one that ensures the process’s global existence and uniqueness without any impact on the exit-time result. We can also relax Assumptions A-2. Indeed, the domain G does not necessarily have to be bounded, but it is necessary to add another assumption on the level sets instead. Consider: Assumptions A-2′. Under these assumptions we can obtain Kramers’ type law and the exit location result. Acknowledgments : This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). Acknowledgments This work has been supported by the French ANR grant METANOLIN (ANR-19-CE40-0009). References [AdMKT23] Ashot Aleksian, Pierre del Moral, Aline Kurtzmann, and Julian Tugaut. Self-interacting diffusions: long-time behaviour and exitproblem in the convex case. preprint, March 2023. [BLR02] Michel Benaïm, Michel Ledoux, and Olivier Raimond. Selfinteracting diffusions. Probab. Theory Related Fields, 122(1):1–41, 2002. [BR03] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. II. Convergence in law. Ann. Inst. H. Poincaré Probab. Statist., 39(6):1043–1055, 2003. [BR05] Michel Benaïm and Olivier Raimond. Self-interacting diffusions. III. Symmetric interactions. Ann. Probab., 33(5):1717–1759, 2005. [CGM08] P. Cattiaux, A. Guillin, and F. Malrieu. Probabilistic approach for granular media equations in the non-uniformly convex case. Probab. Theory Related Fields, 140(1-2):19–40, 2008. [DR92] R. T. Durrett and L. C. G. Rogers. Asymptotic behavior of Brownian polymers. Probab. Theory Related Fields, 92(3):337–349, 1992. [DZ10] Amir Dembo and Ofer Zeitouni. Large deviations techniques and applications, volume 38 of Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2010. Corrected reprint of the second (1998) edition. [FW12] Mark I. Freidlin and Alexander D. Wentzell. Random perturbations of dynamical systems, volume 260 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, third edition, 2012. Translated from the 1979 Russian original by Joseph Szücs. [HIP08] Samuel Herrmann, Peter Imkeller, and Dierk Peithmann. Large deviations and a Kramers’ type law for self-stabilizing diffusions. Ann. Appl. Probab., 18(4):1379–1423, 2008. [KK12] Victor Kleptsyn and Aline Kurtzmann. Ergodicity of selfattracting motion. Electron. J. Probab., 17:no. 50, 37, 2012. [Kur10] Aline Kurtzmann. The ODE method for some self-interacting diffusions on R d . Ann. Inst. Henri Poincaré Probab. Stat., 46(3):618– 643, 2010. [NRW87] J. R. Norris, L. C. G. Rogers, and David Williams. Self-avoiding random walk: a Brownian motion model with local time drift. Probab. Theory Related Fields, 74(2):271–287, 1987. [RW00] L. C. G. Rogers and David Williams. Diffusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition. [TTV12] Pierre Tarrès, Bálint Tóth, and Benedek Valkó. Diffusivity bounds for 1D Brownian polymers. Ann. Probab., 40(2):695–713, 2012. [Tug12] Julian Tugaut. Exit problem of McKean-Vlasov diffusions in convex landscapes. Electron. J. Probab., 17:no. 76, 26, 2012. [Tug16] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions. Electron. Commun. Probab., 21:Paper No. 11, 7, 2016. [Tug18] Julian Tugaut. Exit-problem of McKean-Vlasov diffusions in double-well landscape. J. Theoret. Probab., 31(2):1013–1023, 2018. [Tug19] Julian Tugaut. A simple proof of a Kramers’ type law for selfstabilizing diffusions in double-wells landscape. ALEA Lat. Am. J. Probab. Math. Stat., 16(1):389–398, 2019. [VF70] A. D. Ventcel’ and M. I. Freıdlin. Small random perturbations of dynamical systems. Uspehi Mat. Nauk, 25(1 (151)):3–55, 1970. [Vil09] Cédric Villani. Optimal transport, volume 338 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2009. Old and new. This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

A Generalization of Our Exit-problem for a Class of Non-Markov Processes With Path Dependency Paper

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof. The proof of the following lemma follow the same logic as the one of Lemma 3.3. We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3.3 Proofs of auxiliary lemmas 3.3.1 Proof of Lemma 3.1: Initial descent to the point of attraction 3.3.2 Proof of Lemma 3.2: Convergence of deterministic process towards a Then, by Grönwall’s inequality: That guarantees exponentially fast convergence towards 0: .3.3.3 Proof of Lemma 3.3: Attraction of stochastic process towards a And this concludes the proof of the lemma. 3.3.4 Proof of Lemma 3.4: Behaviour in the annulus between Bρ(a) and ∂G 3.3.5 Proof of Lemma 3.5: Stabilization of the occupation measure Proof . The proof of the following lemma follow the same logic as the one of Lemma 3.3. Proof We have to show now that This completes the proof. Now we are ready to prove Lemma 3.5 itself. That gives us: We recall that Lemma 3.9 establishes the following asymptotic behaviour for small σ: Let us first deal with the term B. By (3.22), we have Consider now A. Separate each probability inside the sum the following way 3.3.6 Proof of Lemma 3.6: Exit before nearing a First, we show that it is possible to establish a lower bound for By Lemma 2.5, which proves the lemma by taking ρ → 0. 3.3.7 Proof of Lemma 3.7: Control of dynamics for small time intervals Use the Jensen’s inequality, integrate the second part of the last integral over s, and get: We apply Grönwall’s inequality and get: Thus, it follows that, for the absolute value of the Brownian motion itself, Taking into account these remarks, we can now use the Schilder theorem [DZ10, Lemma 5.2.1] that provides the LDP for the path of the Brownian motion. Hence, for some fixed ϵ and 0 < T < 1: where d is the dimension of the space. Note that for any ϵ > 0 the following limit holds: 3.3.8 Proof of Lemma 3.8: Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proofs of Auxiliary Lemmas

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3.2 Proof of the main theorem 3.2.1 Kramers’ law Given the results of Lemmas 3.2 and 3.5, we expect our process to spend most of its time near a with σ small enough. In order to have more information about this behaviour, we introduce the following stopping times That gives us: Therefore, by induction in k, we get: By Markov’s inequality, we get: We remark that by Lemmas 3.1, 3.5 and derivations above, we have This concludes the proof. 3.2.2 Exit-location This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: Proof of the Main Theorem

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3.1 Auxiliary results 3.1.1 Initial descent to the point of attraction 3.1.2 Convergence of deterministic process towards a 3.1.3 Attraction of stochastic process towards a 3.1.4 Behaviour in the annulus between Bρ(a) and ∂G 3.1.5 Stabilization of the occupation measure 3.1.6 Exit before nearing a 3.1.7 Control of dynamics for small time intervals 3.1.8 Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3.1 Auxiliary results 3.1.1 Initial descent to the point of attraction 3.1.2 Convergence of deterministic process towards a 3.1.3 Attraction of stochastic process towards a 3.1.4 Behaviour in the annulus between Bρ(a) and ∂G 3.1.5 Stabilization of the occupation measure 3.1.6 Exit before nearing a 3.1.7 Control of dynamics for small time intervals 3.1.8 Uniform lower bound for probability of exit from G This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: The Auxiliary Results

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 3 Exit-time First, we present the following assumptions on domain G and geometry of the potentials that drive the system inside the domain. Assumptions A-2. Now, let us remind the main theorem before proving it. Under the sets of Assumptions A-1 and A-2 the following theorem holds: This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 3 Exit-time First, we present the following assumptions on domain G and geometry of the potentials that drive the system inside the domain. Assumptions A-2. Assumptions A-2. Now, let us remind the main theorem before proving it. Under the sets of Assumptions A-1 and A-2 the following theorem holds: This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

Exit-Problem for a Class of Non-Markov Processes With Path Dependency: A Peek at the Exit-Time

Table of Links Abstract and 1 Introduction 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.4 List of notations 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.3 Compactness results 3 Exit-time 3.1 Auxiliary results 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 4 Generalization and References 2.3 Compactness results This completes the proof. This paper is available on arxiv under CC BY-SA 4.0 DEED license. Authors:
(1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France;
(2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France;
(3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Table of Links Abstract and 1 Introduction Abstract and 1 Introduction 1.1 State of the art 1.1 State of the art 1.2 Some remarks on dynamics and initial condition 1.2 Some remarks on dynamics and initial condition 1.3 Outline of the paper 1.3 Outline of the paper 1.4 List of notations 1.4 List of notations 2 Large Deviation Principle 2 Large Deviation Principle 2.1 Establishing the LDP for the SID 2.1 Establishing the LDP for the SID 2.2 Results related to the LDP 2.2 Results related to the LDP 2.3 Compactness results 2.3 Compactness results 3 Exit-time 3 Exit-time 3.1 Auxiliary results 3.1 Auxiliary results 3.2 Proof of the main theorem 3.2 Proof of the main theorem 3.3 Proofs of auxiliary lemmas 3.3 Proofs of auxiliary lemmas 4 Generalization and References 4 Generalization and References 2.3 Compactness results This completes the proof. This paper is available on arxiv under CC BY-SA 4.0 DEED license. This paper is available on arxiv under CC BY-SA 4.0 DEED license. available on arxiv Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France. Authors: Authors: (1) Ashot Aleksian, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France; (2) Aline Kurtzmann, Université de Lorraine, CNRS, Institut Elie Cartan de Lorraine UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France; (3) Julian Tugaut, Université Jean Monnet, Institut Camille Jordan, 23, rue du docteur Paul Michelon, CS 82301, 42023 Saint-Étienne Cedex 2, France.

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