You could start with the first chapter of this book, and then with this three-volumes book. The former is a very nice mathematical introduction to finance, from the viewpoint of someone on the mathematical (or physical) side. The latter may seem, and is, a book on interest rates, but it allows you to cover all mathematical techniques used in finance nowadays, and its first volume is the best introduction I have ever seen on mathematical finance ; it has btw a very nice bibliography that will redirect you to central papers in the discipline etc. I am not that fan of this book, even if I started in the field with him, but it could be ok nevertheless for what you are looking for. Finally, there is a book that is not very good on mathematical finance at all, but it is central on FX implied volatility quoting conventions, and is a must have for this.
Last point, previous books are not books on stochastic processes or PDE's or other mathematical subjects that are used in mathematical finance, they are books on mathematical finance roughly covering these subjects, and using and applying them to fianance - essentially pricing and hedging, curve building etc. This means that sometimes you will need to put your nose in a book or another on stochastic processes or even probabilities (note that this book on probabilities and discrete time martingales is a must-have), or PDE's etc. Theses are my complementary advises. I know that this wasn't you primary question, but I don't see myself giving a piece of advise on mathematical finance without mentioning this.
Ultimately, this is a very broad question so I won't even attempt to answer it completely. Stochastic PDEs are entire area of active research. Your confusion basically boils down to "what are SPDEs" which people spend careers answering.
Let me briefly remark on a few points:
On the LHS we have the Laplacian operator, which is the divergence of the gradient. What does a PDE with this operator imply about the solution?
If you have studied (non stochastic) PDEs you should have studied the Laplace equation, $\Delta u =0$, or with driving force $\Delta u = f$. I couldn't write up a full treatment of Laplace's equation so I won't. In most SPDEs there is a time component so there are the heat equation, $\frac{\partial u}{\partial t} =\alpha \Delta u$, and Schrödinger equation, $\frac{\partial u}{\partial t} = (i\alpha \Delta +V)u$. I can't go through a full treatment of these, either.
What you have is not exactly the Laplacian operator. You have a generalization called the fractional Laplacian operator. We define this operator by its Fourier transform. Recall: $\mathcal{F}(\Delta u )(\textbf{k})=\|\textbf{k}\|^2\mathcal{F}(u)(\textbf{k})$. So the so called "fractional Laplacian operator" has the following property: $\mathcal{F}(\Delta^{\alpha/2} u )(\textbf{k})=\|\textbf{k}\|^{\alpha}\mathcal{F}(u)(\textbf{k})$. See their definition of $(\kappa^2-\Delta)^{\alpha/2}$ here, which should not be that surprising.
On the RHS we have a stochastic white noise process W. How is this different from putting something deterministic here? In the paper they call this "driving the SPDE with white noise" but I don't know what driving means in this context.
At the heart of SPDEs is that noise term, that is the object of study. $\mathcal{W}$ is a distribution, meaning it is an object like a Dirac $\delta$ "function". It is a Gaussian distributed random field. Defining this object rigorously is not trivial. Like the Dirac $\delta$, it only makes sense when paired with nice functions (this is the integral formulation I was telling you about in the comments. This is very important). I recommend this paper by Davar Khoshnevisan, pages 1-4 (including equation 2.1) for a fairly complete definition of this process in the case of the stochastic heat equation which should give some insight. Also see this question for more information on the $\delta$ "function".
Here is a video of Field's medalist Martin Hairer explaining to a layman what and why we should study stochastic PDEs (this video is absolutely painless, no real math). This should answer your questions.
They mention in the paper the relationship of this equation to diffusion. It would be helpful to flesh out that connection.
I actually did my undergrad thesis on this connection. Basically, the (non stochastic) heat equation can be derived and solved entirely in terms of Brownian motion. Einstein did this in 1905, from a macroscopic level and Smulochowski did this from a microscopic level. I go over both derivations in my thesis. You can solve the heat equation in terms of expectation of Brownian motion. This is called the Feynman Kac formula. If you do Wick rotation to imaginary time, you get the solution to the Schrödinger equation. The solution to the Schrödinger equation is given by Brownian motion running in imaginary time. This is known as Feynman path integration, and was the subject of his PhD dissertation.
See also a post I answered on MO for this connection.
How does this relate to the intuitive picture from the simpler equation?
I'm not sure, I haven't read their paper, sorry. :)
SPDEs are a beautiful and useful field, I am glad you came across it. It is, however quite technical. I recommend a few mathematician's papers. I recommend:
Samy Tindel
Carl Mueller
Davar Khoshnevisan
I want to remark on one last thing. The field of SPDEs was recently changed by Martin Hairer's work on rough path theory and regularity structures, for which he received the Field's medal in 2014. I asked a question about this when I first joined math.SE, you can see here. I have since learned a great deal about rough path theory. You can see Martin Hairer's papers here, but be forewarned that they are extremely advanced.
I'm not sure if I answered any of your questions satisfactorily, but I hope I at least gave you some material to look for the answers. I am glad you are interested in this field!
If this is too much information, just watch the video from Martin Hairer. It is painless.
Best Answer
Very roughly speaking, you can think of a stochastic process as a process that evolves in a random way. The randomness can be involved in when the process evolves, and also how it evolves.
A very simple example of a stochastic process is the decay of a radioactive sample (with only one parent and one daughter product). Initially, it has some large number $N$ of atoms of the parent element. Over time, the number of such atoms decreases, always by $1$, but at random moments in time. The state of the system can be represented by $k$, the number of atoms of the parent element present at a given moment in time. Initially, $k = N$, but eventually, it will fall to zero.
In this process, when the state changes is random, but not how it changes. In other processes, such as a discrete-time random walk, when the state changes is deterministic, but how it changes is random. And there are other processes in which both when the state changes and how it changes are random.
Interestingly, in many cases, stochastic processes are used to model situations that may not have inherent randomness. For instance, Brownian motion is the result of forces that could, in principle, be determined precisely (if we ignored quantum mechanics). However, the number of objects in a normal system is so large that such an analysis would be intractable. Instead, we model the motion of objects using a stochastic process, and thereby obtain some insight into the behavior of such systems (for instance, the statistical behavior of a given particle over time) that we could not begin to with a deterministic approach.