 Thank you very much to the organizers for this beautiful day. So I will be talking to you about some results I found on Bessel processes, but the initial motivation was in studying some solutions to stochastic PDEs, or SDEs with the drift which is singular. So here is the motivation, so we have solutions to some complicated equations and we're interested in some properties of the corresponding semi-group. In particular we want to know if we have some continuity of the semi-group with respect to the initial condition which translates formally in the notion of strong-feller property which I'm going to explicit later on. So my plan is the following, I'm first going to explain what the strong-feller property is, then I will explain to you a very efficient tool to prove strong-feller bounds which is the Bismuth-Edward-Ely formula and in the last part I will explain to you how one can obtain such a formula in a very non-dissipative setting which is given by the Bessel processes. So the setting is the following, consider any Polish space E and Markovian semi-group PT over E. So I will be considering two test, two spaces of test functions, the space of Borel and bounded functions in E which I've been denoted by B sub B and the space of continuous and bounded functions in E. Now the semi-group PT said to have the fellow property E for all t and all phi which is continuous, PT of phi would also be continuous and bounded. So this is a property which holds in very general cases but the property that I will be interested in here is the strong-feller property which says that PT of phi will be continuous for any phi which is only Borel. So it may be very discontinuous I priori but for any time t which is positive PT of phi should be continuous. So it is strong-feller if the semi-group creates continuity in positive time. So in the sequel I will focus on the case where E is given by the real line and then the semi-group is given by expectation of phi of xt of x where xt of x is the solution of some SDE with drift given by B and diffusion coefficient given by sigma which I suppose to be smooth functions on R and initial conditions given by x. And so here are two examples so first consider the case where sigma is 0 and B is equal to 1 then we are just in the case of an ODE and in that case clearly the semi-group is just given by translation of the initial condition by t and obviously it is not strong-feller because it does not create any regularity of the function. On the other hand consider the case where B is 0 and sigma is equal to 1 so it is the case of random motion then in that case the semi-group is given by convolution by the heat kernel which is given here and this is a smooth kernel so pt is not only strong-feller but even better pt of phi will be smooth for any phi which is Borel for any positive t. So here are two cases that are very different and we see that the important fact is that in our SDE here the stochastic part should be non-degenerate enough for the solution to have some smoothing property here and so the PDE interpretation of that is that in the first example I gave to you we have a hyperbolic PDE so a transport equation whereas in the second example we have an elliptic PDE which creates regularity and what happens in general coefficients? Well we have very sharp criteria for example Hermander's criterion which says that if sigma is sufficiently non-degenerate in some sense then for all t strictly positive pt of phi will be smooth for any phi which is Borel so in particular strong-feller property holds but what I'm interested in is for some quantitative versions of the strong-feller property my problem is for any given t to quantify the difference pt phi of x minus pt phi of y when x and y are close. So the question is how to obtain a bound of this quantity. So the very efficient tool to do that is the bismutal worthily formula I'll explain to you now. So to understand this formula we'll suppose that sigma is equal to one here but it's not necessary just to simplify the notations and I suppose that B satisfies Lipschitz bound with the constant c but the important assumption I'll make is that B satisfies this one-sided bound of the derivative that is B prime is uniformly bounded by some constant l which is a real number. Now the idea of the bismutal worthily formula is to look at the dependence of the solution of the sd with respect to the initial condition. So consider the derivative of the solution with respect to the initial condition that I'll call eta then by differentiating the sd with respect to x it is not difficult to see that eta t solves the variational equation given by this differential equation. So this is an ODE with an explicit solution eta t of x and we remark that it is first it is non-negative which shows that x of t is an increasing function of x and moreover it is bounded by e to the power l t. Why? Because I have made the assumption here that B prime is bounded by l. So this gives us boundedness of the derivative eta. So what we can say is that for phi which is c1 bounded by the dominated convergence theorem the derivative of the semi-group PTF phi is given by this formula here. But the problem here is that we have a derivative here and we would like to get rid of the derivative because we want a strong fellow property so we should have an expression here which only involves phi instead of phi prime and the way of getting rid of this derivative is by using the integration by part formula and that's the idea of the Bismut-Elwetely formula so we have the following equality the derivative of the semi-group PTF phi of x is given by 1 over t expectation of phi of x t of x and multiplied by this stochastic integral integral from 0 to t of eta s dBS so here as you can see we got rid of the phi prime here we have no more derivative but there's the cost that we have this new random variable here which we have to bound. So the corollary of this theorem is that we have the following strong fellow bound because if we integrate this formula here we obtain that for any x and y we have a bound in the supreme norm of phi and we have this constant e to the power L where L was the bound on B prime and we have 1 over square root of t and we have a global Lipschitz bound in x and y so that's a very nice answer to that question here in a nice setting and the crucial remark is that this bound here only involves L it does not involves the Lipschitz constant on the drift coefficient B and that's the reason why this Bismuth-Arwathili formula is very interesting and very successful for example when studying more complicated systems as SPDEs so the history of this formula started by formula particular form of this formula discovered by Jean-Michel Bismuth in 1984 and 10 years later the formula as I presented it to you was derived by David Alworthy and Shremé Lee and so they proved this formula and also gave variance for higher order derivatives for the semi-group of solutions of SDEs on some manifolds and later on there were other so there were generalizations of this formula but all these generalizations were made on the assumptions mainly that as I said above B' was bounded now the question is what happens when B' is not bounded above by some constant L because there are very important SPDEs where this assumption does not hold so last year there were two articles about some semi-linear SPDEs that have very exotic drift term B that is hardly dissipative so this means that the bound that I gave above was really not satisfied and so these authors proved that nevertheless there was a strong fellow property although they did not prove Bismuth-Alworthy formula but they used methods close in spirit to the Bismuth-Alworthy formula but now these articles did not answer the problems encountered when I studied SPDEs as the following so I'm interested in evolution SPDEs with a drift term in 1 over U-Cube so when C is positive we have no problem actually my PhD advisor Lorenzo Zambotti proved well-posedness for these SPDEs when the drift term is of the form C over U-Cube with C positive and he proved that also we have strong fellow property and this relied on the fact that the function C over U-Cube is decreasing on R plus so we have some dissipativity and the question was what happens if C is negative because in this case we don't have dissipativity anymore so this problem is very difficult and I still don't have any solution so I said let's look at some one-dimensional example which has similar difficulties but which is easier to handle this example is the example of Bessel processes so Bessel processes are the following processes so it's non-negative processes with real value which satisfies this SDE with a drift term in 1 over X and we require that the 0.0 is instantaneously reflecting so it turns out that there is a unique for all delta which is positive there is a unique solution to this equation which is a delta-dimensional Bessel process and so a remark is that if delta is an integer this process has the same law as the Euclidean norm of a delta-dimensional Brownian motion and so when delta is bigger than 1 here the drift is dissipative so actually it is decreasing so we have no problem we have existence and unicity result but the problem comes when delta is smaller than 1 and to obtain well-posedness of this equation we actually resort to some transformation of the square and we see that the square satisfies some well-posed SDE this is the way we obtain the well-posedness for this SDE for any delta and okay so the question is do we have a strong follower property for the Bessel semi-group so in order to answer this question we can use we have an explicit formula of the semi-group with a density with respect to Lebesgue measure on r plus and so we can compute the derivative of the semi-group with respect to X and we obtain the following so for all t positive the semi-group of the delta-Bessel is differentiable on r plus and the derivative involves so it's X over t times the difference of the semi-group of the delta plus 2 Bessel minus the delta Bessel so there's a nice interplay between the delta Bessel and the delta plus 2 Bessel and as a consequence we have a strong follower bound here which is given by Lebesgue so we have a Lebesgue bound here Y minus X which is not global it is local but yet it is very nice formula and my question is can we interpret this equality as a Bismuth-Edouard-Hilly formula so the answer is yes but it's really not easy to see why? because as I said we don't have dissipativity when delta is smaller than one so for example if we compute the derivative of the solution with respect to initial condition eta t here so remember here the drift what I called B is given by delta minus 1 over 2 X so the derivative is given by 1 minus delta over 2 X2 so you see that the supremum over r plus star of B prime so it's zero if delta is bigger than one but it's plus infinity if delta is smaller than one so we see that when delta becomes smaller than one we get into trouble and this can be seen when we look at the derivative of rho because we have this we can see that when delta is smaller than one eta t will go to infinity as t goes to t0 where t0 is the first hitting time of the origin nevertheless we have a nice proposition which is a variant of results of by Pitman Yor which says that the delta plus 2 Bessel is absolutely continuous with respect to the delta Bessel on any finite interval and we have an expression for this Radon-Nikodym derivative and okay then we can use some nice formula on rho and eta I will not detail here and as a consequence we finally obtain this Bismuth-Arworthy formula which holds for any delta positive so it looks like the formula we had before but the difference is that here this martingale is not in L2 actually it will be an LP martingale for some p depending on delta which may be smaller than 2 and I can obtain a better bound on the better Lipschitz bound than before I can improve them using this representation here I can improve the exponent on t I had before so I can obtain an exponent 1 over t to the power alpha of delta where alpha of delta is smaller than 1 depending on so its value which depends on delta so okay so the conclusion is that we obtain the Bismuth-Arworthy formula in a very non-dissipative case so which is very surprising and so okay there is something not very satisfactory in the fact that we use very particular properties of the Bessel processes but so we should try to develop more conceptual tools to treat more general processes but we have some partial results that suggest that at least the strong feller bound of the type I've shown still hold for more general systems and that's all and I have here some references for you and I thank you for your attention Thank you Is there a question? Henry, do you have a question? A question for me A question? No Yes, do you have other applications in mind besides showing that these are feller? The Bismuth-Arworthy formula Yes In fact, why am I speaking in French? Because you can speak in English So the application is Are we interested in strong feller property? So I didn't mention that but for example if you have the strong feller property Actually, that was not my question My question was do you have other applications in mind? For SPDEs, for example Yes, so we would like, for example to use similar to find similar results for SPDEs of the kind that was here when C is smaller than zero So in that case here this is the function C over X cubed will be increasing so we do not have any strong feller results yet and actually we don't have well-posed results of such SPDEs So these SPDEs are interesting because they have some scaling properties but we still don't know how to define them when C is negative Another question How to make a small informative change for questions Thank you Henri